help-49

[midnight plank]

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[gilded raptor]

I don't really understand this function

Anybody could give me some tips and pointers on what to do?

[prime hornet]

what’s the problem at hand?

this is a piecewise defined function

are you being asked to find its value at x = 2, or determine its continuity?

[gilded raptor]

It's says here "Analyze the following functions and find their parameters"

Sorry, English is not my native language -

[wanton spade]

sorry no other messages had loaded apart from the first

[gilded raptor]

It's fine, don't worry about it

Im quite literally stuck and the first assignment and really confused

[wanton spade]

not sure what the question is asking exactly

maybe send the image of the original question

someone else will probably be able to figure it out

[gilded raptor]

Oh yeah, but it's in Spanish btw

Wait, gimme a sec

Analyze the graphs of the following functions and indicate their parameters (Parameters: Domain, Range, Roots, y-intercepts, Intervals of Increase and Decrease, Sets of Positivity and Negativity)

I just would like an explanation of How to do it, I really don't get it

[wanton spade]

ohh

do you know what each parameter means?

[gilded raptor]

Blue dot for transparency

[gilded raptor]

Domain and Range yeah, I don't really get roots tho

[wanton spade]

roots is basically where the function is 0

i assume you can find the domain and range

[gilded raptor]

Yeah

[wanton spade]

where is the functions in the graph zero?

[midnight plank]

@gilded raptor Has your question been resolved?

[gilded raptor]

Oh yeah, btw I just encountered my teacher and he's helping thank y'all sorry for the trouble

[midnight plank]

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[twin kernel]

fx=y(4-2x-y)=0 and fy=x(4-x-2y)=0

idk what after

[lyric charm]

well (0,0) is an obvious one right

[twin kernel]

how i find the others

[midnight plank]

@twin kernel Has your question been resolved?

[midnight plank]

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[glossy spindle]

Please can someone give me a clue how to start

[dreamy lichen]

for the 15(a) use the sin(2x) identity

[glossy spindle]

2sinxcosx

[dreamy lichen]

yep, that

then factor

and for b, a simple substitution will reduce it to a

[glossy spindle]

Huh?

What do i do after?

[dreamy lichen]

are you doing a now?

or what

whats your current progress

[dreamy lichen]

okay, now try factoring it

[glossy spindle]

Take a common factor of 2cosx?

[dreamy lichen]

exactly

there is 0 on the right side, so factoring is really useful here

[glossy spindle]

Let it =0 too?

[dreamy lichen]

i mean you are solving sin(2x) + 6cos(x) = 0

[glossy spindle]

So 2cosx=0

[dreamy lichen]

so ofc it's = 0

[glossy spindle]

And sinx=-3

[dreamy lichen]

yes

[midnight plank]

@glossy spindle Has your question been resolved?

[midnight plank]

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[midnight plank]

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[paper prism]

. This solution is rejected though because it is not possible

[midnight plank]

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[fair crest]

Trying to get some late night reps in. Am I working this problem down correctly? I know I have to get this problem down into the quadratic equation. Once I get it there correctly I get the question right 100% of the time. I just struggle to actually get it there

[pearl mantle]

The quadratic formula actually overkills this question.

Since you're solving for y.

From line one, you move everything to the Right Hand Side.

In which you'll obtain $7y^2 = 3 + x - 2x^2$.

[grand pond]

Restarter

[pearl mantle]

Then you can divide both sides by 7.

And Square Root both sides.

Rationalise the denominator if required.

[fair crest]

Okay, so looking back with your notes.

1. I messed up when I was moving things over to the right hand side. I ended up with a negative 3 because I put it through the same procedure as the numbers on the left. (“Moving it “unnecessarily )

2. While I’m right that -7 doesn’t belong with Y. I’ve actually over complicated what I need to do. I divide both sides to get rid of 7 and then

Can you explain why I’m square rooting please?

I’m going to put the things I’m typing in this help channel into my notebook, so I can reread and tell myself why I got this problem wrong. Then tell myself the correct things to do

[wooden badger]

bcz we r solving for y not y²

[fair crest]

Okay now, if this problem were asking for y^2 what would I be doing in that case?

[wooden badger]

if tht was the case, this could be ur final ans

after dividing both sides by 7

[fair crest]

OHHHH

GOT IT

Thank you both for your help. This was clear and concise. ❤️

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[twilit field]

can I have a hitn

[gaunt jetty]

Hint 1: ||Recall the definition of delta(S)||
Hint 2: ||Break it into cases||

Hint 3: ||Two of the cases are trivial; there is only one you have to worry about||

[twilit field]

I tried hint 1 and hint 3 earlier 😭

[unkempt sluice]

What is δ?

[gaunt jetty]

Hint 3 relies on Hint 2 so idk what you mean

[unkempt sluice]

Diameter?

[gaunt jetty]

the supremum of the distances between any two points in that set.

[unkempt sluice]

Triangle inequality

[gaunt jetty]

That was going to be my Hint 4

[unkempt sluice]

d(x,y)<=d(x,z)+d(z,y) thing, choose z in the intersection

Oh

[twilit field]

😔

how did I miss that

got it

thanks

.close

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[inland patio]

If $X_1,\ldots,X_n$ are independent, are then $t^{X_1},\ldots,t^{X_n}$ also independent? I know the collection ${f(X_i)}$ for some measurable function $f$ would be independent because these are all $\sigma(X_i)$-measurable, but I'm not sure what the function $f$ is in my question. The context is probability generating functions.

[grand pond]

psie

[lyric charm]

well the function x \mapsto t^x is measurable innit

[inland patio]

Yes. So you fix t?

[inland patio]

Or do you perhaps mean (t,x) \mapsto t^x?

[lyric charm]

that's what i understood you to mean

[midnight plank]

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[wooden badger]

A straight line passes through (8,2) and cuts the positive co-ordinate axes at P and Q. Find the minimum value of |OP| + |0Q| where O is the origin.

[lusty python]

wdym positive coordinate axes?

[inner barn]

perhaps, look at a line with varying gradient/slope and then see how the intercepts change according to this

[wooden badger]

positive x and y axis

[next bobcat]

negative slope then?

yeah

[wooden badger]

yep

[next bobcat]

oh Kay

I think drawing the graph is the most important first step

[wooden badger]

i did

[bold peak]

Have you covered differentiation

[wooden badger]

i wrote the eqn of the line aswell

but idk wht to do after tht

[wooden badger]

a lil bit.

[bold peak]

Are you allowed to use it for this problem

[wooden badger]

yes

[bold peak]

Cool

[inner barn]

okay, now, look at the quantities you are trying to minimize

can you write them in terms of what you know? how do you minimize this?

[bold peak]

Find the x intercept in terms of the y intercept

Or the other way around

[next bobcat]

that’s what I thought

I think you should find like the bare minimum x and y values if you can’t do diff.

[wooden badger]

the only relation i could get is 8Q+2P=PQ

[next bobcat]

ok

So OP, OQ and PQ form a rat

[wooden badger]

rat?

[orchid saddle]

right angle triangle

[wooden badger]

oh yes

[next bobcat]

sorry I was distracted

ok so

[inner barn]

. . .

[next bobcat]

One sec I need to draw a diagram myself

[inner barn]

what was your equation of the line?

[next bobcat]

I think this is the one?

where p is the x intecept and Q is the y intercept

so 8y + 2x = xy

[inner barn]

?

that is not a line

[next bobcat]

that’s the equation op is talking about I think

[next bobcat]

there are multiple lines that you can get with what we have

[inner barn]

?

[wooden badger]

x/P + y/Q = 1

[inner barn]

okay wait

ch3rry do you use the term gradient or slope

[next bobcat]

Ok

[wooden badger]

slope yes, gradient no.

[inner barn]

ok

suppose the slope is m, do you remember what the general equation of a line passing through a point is

[wooden badger]

i do

[lusty python]

spit it out

[wooden badger]

y=mx+c?

[inner barn]

through a point

like if we pass through a point (p,q), do you know the equation of the line given that the slope is m

[wooden badger]

one point form?

[inner barn]

yes

[wooden badger]

m=(y-p)/(x-q)

oh i interchanged p and q :,)

[inner barn]

yeah

[lusty python]

...

[inner barn]

but apart from that, it is ok

[wooden badger]

yep

[inner barn]

now, from this information we can find your P and Q in terms of m

why are we doing this? because we are trying to minimize |OP| + |OQ|, so if we have it in terms of one variable, it is easy

[wooden badger]

wht abt intercept form ,since theyre given

[inner barn]

?

[slender walrus]

your choice of using intercepts form was fine

and is what i would've used

[inner barn]

right but the problem is that you don't know the intercepts yet

[slender walrus]

you have expressions for them and a relation for them

[wooden badger]

wht is the quotient rule for finding the derivative/

[slender walrus]

you're trying to optimise

minimize |OP| + |OQ|
and since P and Q are positive,
that'll simply be P + Q

[inner barn]

wait ok ch3rry what step are you on

[wooden badger]

lemme try minimizing using calc

[slender walrus]

and you can then use

8Q+2P=PQ

to express that in terms of P or Q depending on personal preference

[inner barn]

okay, i will let you finish helping then since maybe your method will make more sense to her

[wooden badger]

okay i got my ans by minimizing the func

18

[inner barn]

yes, i got the same

[wooden badger]

but we havent studied derivatives yet

so any other methods ygs hv?

[late rover]

There's another way

[inner barn]

?

[late rover]

yk intersection form of a line?

x/a+y/b=1

[slender walrus]

that's what she used at the start

[late rover]

Oh okay

so, 8/a + 2/b =1 and use titu lemma inequality

we have 8/a + 2/b = 1 >= 18/{a+b}

done

[wooden badger]

wut😭

[late rover]

that's it, really

[wooden badger]

hvnt studied tht~~~

[late rover]

You can derive titu's lemma from AM-GM

[wooden badger]

youd need to teach me TT

[late rover]

I kinda forget how to do so, gimme a min

[wooden badger]

k

[inner barn]

i think, this is not very informative

[wooden badger]

AM and HM inequality can also be used

[inner barn]

unfortunately, my syllabus was quite different back in highschool so i'm unsure what method the others were trying to get at
but maybe instead of pulling out a random inequality you haven't learned, you should see if you can follow their lead instead

[orchid saddle]

I think we overcomplicated a simple question here

[inner barn]

the question has already been solved

but she is looking for a different method

[orchid saddle]

Oh I see

[wooden badger]

can someone explain this one?

[orchid saddle]

No idea what that is

[wooden badger]

my teacher used weighted mean

[late rover]

okay found the proof

Kinda weird with AM-GM tho, people usually use Cauchy-Schwarz to prove it

It also has a different name which is Engel's form of Cauchy-Schwarz

so, (x^2/a + y^2/b )(a+b)

expand this

We'll get x^2+y^2 + bx^2/a + ay^2/b

Use AM-GM

bx^2/a + ay^2/b >= 2xy

Hence x^2+y^2 + bx^2/a + ay^2/b >= x^2+y^2 + 2xy = (x+y)^2

or

(x^2/a + y^2/b )(a+b) >= (x+y)^2

therefore x^2/a + y^2/b >= (x+y)^2/(a+b)

This's titu's lemma

[wooden badger]

which nos did u use am gm on?

[late rover]

bx^2/a + ay^2/b >= 2xy

[wooden badger]

ohk

[lethal path]

doesn't Titu's give 8/a + 2/b >= (sqrt8 + sqrt2)^2 / (a + b) though?

[late rover]

yeah

,w (sqrt8 + sqrt2)^2

[grand pond]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by fionna_fx

[lethal path]

oh that's sick

right ofc

[wooden badger]

ohhhhhhhhhhhhh i got itt

i still need to know how am hm inequality with weighted means can be usd here

[late rover]

remind me what AM-HM is again

[wooden badger]

arithmetic and harmonic means

[late rover]

I mean the formula...

[wooden badger]

oh lmao

for 2 nos?

[late rover]

yeah, tbh I barely use Am-HM

[wooden badger]

am>=hm..hm=2ab(a+b)

[late rover]

ehh

1 = 8/a + 2/b = 4/a + 4/a + 1/b >= 9/(a/4 + a/4 + b/2) = 9/(a/2 + b/2)

[late rover]

Like this ig

I had never used AM-HM before, this's the first time and it's pretty weird lol

Looks right

It's a+b >= 18 also

[wooden badger]

i need u to explain TT

[late rover]

hmm I thought it's pretty obvious

So

8/a + 2/b = 1 right

[wooden badger]

mmhm

[late rover]

inequality with weighted is a technique where you split the term into more terms in order to obtain the extrema we desire to find

what we want: a+b

[wooden badger]

ok....

[late rover]

Okay this's hard to explain 🥀

[wooden badger]

[late rover]

if we apply AM-HM directly, we get 8/a + 2/b > 9/(a/8 + b/2)

but we can't obtain a+b from a/8 + b/2

Notice that if I divide 8/a into 4/a + 4/a

[wooden badger]

ok.........

[late rover]

Then if we apply AM-Hm 4/a + 4/a + 2/b > 9/(a/4 + a/4 + b/2) = 9/(a/2 + b/2)

= 9/(a/2 + b/2) = 9/((a+b)/2) = 18/(a + b)

You see, now a+b appears

This's what we want in the first place, to obtain a+b

[wooden badger]

we got the ans

[late rover]

I hope you understand, inequalities are weird ahhhh

[wooden badger]

inequalities are fine

[late rover]

yeah basically

[wooden badger]

i js hvnt worked with weighted means

no way i could've thothought of ts myself

[late rover]

weighted means is a very first technique to deal with inequalities, there're tons of techniques

[wooden badger]

IInequalities As in?

[late rover]

In what you just did, or olympiad type of inequalities

[bold peak]

Are we still on the same question

[late rover]

Yeah, kind of, we're go through other methods to solve the problem

4 so far or 3

[bold peak]

Ah cool, was just wondering if smth needs pinning

[late rover]

you're worry too much,and I have helpful in my different account anyway

[bold peak]

In my defense I have no idea who your other account is

[late rover]

Fair

[bold peak]

Either way, is differentiation one of the methods that was covered

[late rover]

Yeah, I believe so, and it's the the first one to use also

[bold peak]

Cool cool

[junior flower]

xavier is orange now

[bold peak]

It feels weird yeah

[wooden badger]

cool i got 4 methods for the ques now

[late rover]

non of those are geometrical 🥀 sad

[wooden badger]

yikes geometrry

thnx everyone

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[twilit field]

oh, that's where I MESSED UP

nvm

sorry

.close

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[clever sedge]

😂

[twilit field]

I thought 0 was in K^x

😔

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[woeful turret]

how to do this

[silk pulsar]

Yea sure

[wanton spade]

how many times does the function cross the x-axis between 0 and 4?

[woeful turret]

4

[silk pulsar]

Hi

If u don’t mind can i just

Butt in maybe

If i feel like saying smth

[wanton spade]

and f reaches a maximum between each point where the function is zero, right?

[woeful turret]

maximum and minimum right

[wanton spade]

yes sorry

[wanton spade]

what's the derivative at each local maximum and minimum?

[woeful turret]

0

[wanton spade]

so how many times is f' 0

[woeful turret]

3

[wanton spade]

(mean to be clearer, at least 3)

[woeful turret]

ye

[wanton spade]

f' crosses the x axis at least thrice

[woeful turret]

ye

[wanton spade]

so how many times at the least does it attains a max

or min

[woeful turret]

2?

[wanton spade]

so how many zeroes does f'' have at the least

[woeful turret]

2

[wanton spade]

similarly, can you tell me how many zeroes f''' has

[woeful turret]

1

[wanton spade]

yep

now we have to use rolles

we know f''' is zero at least one point c

f'''(c)=0

how would you use rolles to write that in terms of f''

[woeful turret]

uhh im not sure

f''(a)=f''(b)

?

[wanton spade]

shit I've to go I'm really sorry

[woeful turret]

bruhh

[wanton spade]

I'll try to be back in a couple of minutes

[woeful turret]

😭

[wanton spade]

forgive me once again:')

[midnight plank]

@woeful turret Has your question been resolved?

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[night gyro]

hello is my method correct

[inner barn]

not quite

very close, though

the perimeter is 20, but remember that the longest side must be the hypotenuse

[night gyro]

how many marks would i get

[inner barn]

0, unfortunately

[night gyro]

20-4-7

and thats the hypotenous?

[inner barn]

right

so you are actually verifying whether or not 4^2 + 7^2 = 9^2

[night gyro]

if it equals to that it is right angle

if not

is not?

[inner barn]

correct

[night gyro]

ok thanks

[inner barn]

np

[night gyro]

.close

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[prisma egret]

Does a continuous extension of a function have to be unique? Like what exactly is a continuous extension, I find it to be a very loose definition of what it is

[junior flower]

not in general

for certain “extension sets” it may be though

[prisma egret]

I don't know exactly what you mean, what definition do you base that on?

[junior flower]

like an extension by one particular point

that has to be unique if it exists

[fresh sparrow]

https://math.stackexchange.com/a/553347/1153468
have you checked this out? i find it to be a pretty good explanation

[prisma egret]

I have. But it only talks about evitable discontinuities

Is it only for evitable discontinuities?

[fresh sparrow]

yes

[prisma egret]

Then I get it

[fresh sparrow]

if the discontinuity is inevitable, well, it's inevitable, you can't do anything to get rid of it

[prisma egret]

True I was thinking of interpolation

But nevermind

.close

[midnight plank]

Channel closed

Closed by @prisma egret

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

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• Show your work, and if possible, explain where you are stuck.
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• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[rough wasp]

i don't know where to begin

[rough wasp]

<@&286206848099549185>

[verbal sparrow]

Do you know what amplitude means? Like the amplitude of a sin function?

[rough wasp]

I don't know what amplitude means

[verbal sparrow]

On a sin function, its basically the max height of the function

[rough wasp]

the height is infinite though

[verbal sparrow]

Yep, so no amplitude is a valid answer

[rough wasp]

hmm

[verbal sparrow]

What about the period?

[rough wasp]

the period in physics is like time

but i dont know it in math

[verbal sparrow]

Like time, period is "how long" it takes to repeat

In this case, how far along the x-axis does it take to repeat

[rough wasp]

i noticed some limits

[verbal sparrow]

Yep

[rough wasp]

so (-oo, -270) U (-270, -90) U (-90,oo)

[verbal sparrow]

That would be the whole domain of the function, although the bounds toward +/- infinity isnt quite correct

What do you think happens if you were to look further to the right of the graph?

For bigger values of x

Also, I should probably ask, do you know what function this is a graph of?

[rough wasp]

nah

[rough wasp]

i think there might be another limit if we look further

[verbal sparrow]

Its some form of tan(x), not sure if youd know unless you were shown it

[verbal sparrow]

Yes, youre right, vertical limit that is

[rough wasp]

oh yeah, i didn't learn it yet

[verbal sparrow]

It'll repeat like that forever, just like any trig function repeats

So, for the period, bow far does each x value go before it repeats

[rough wasp]

oh okay i see

[rough wasp]

i counted and it spans 180 units

[verbal sparrow]

Yep

Btw, are the answer options text or multiple choice

[rough wasp]

text unfortunately

so for the period, do i just enter 180?

it says for the format of answer to enter an integer, so i just entered 180

[tawdry kraken]

Text inputs in myopenmath is a travesty

[verbal sparrow]

Yeah should be fine, im not sure what it would want for amplitude

[tawdry kraken]

Sometimes they define amplitude directly as the coefficint $a$ in the expression $a\cdot\sin(b(x-c)) + d$

[grand pond]

Coolempire93

[rough wasp]

let me see if theres anything in formula sheet, not sure why i would check that tho

[tawdry kraken]

So whatever comes before your tan should be it

[rough wasp]

lol nothing

[rough wasp]

wait what

[tawdry kraken]

(which is stupid for trig functions other than sin and cos mind you but some people do it)

[verbal sparrow]

And then the function we know is a transformation of tan(x). Normally, tan increases as x increases (before repeating), but its flipped vertically.

[rough wasp]

well tan is decreasing in this case while x is increasing (edited)

how is this related to what we are talking about

[tawdry kraken]

<@&268886789983436800>

[verbal sparrow]

Bot or scam

[solid iris]

both

[verbal sparrow]

Or both lol

[tender trench]

both!

[verbal sparrow]

Yeah so to flip a function across the x-axis, you just negate it

And that should end up being your answer, no extra transformations needed

[rough wasp]

so its just -tan(x)?

but what about the period

[verbal sparrow]

Oh, if you notice its tan(0) = 0, so its already at the origin, it doesnt need to be shifted

[rough wasp]

oh yeah i noticed, thank you ill plug that in now, f(x) = -tan(x)

[verbal sparrow]

And the period of tan(x) is naturally 180, so it doesnt need to be stretched or anything

[rough wasp]

ahh i see, teacher gavve no lessons for chapter 1 saying it was easy lol

[verbal sparrow]

If you have to put a number for amplitude, maybe try 1, since it isnt squashed vertically

[tawdry kraken]

This makes the amplitude 1 by the way (if undefined doesn't work as an asnwer)

What weaboo said

[rough wasp]

can u guys teach me why 1 should be amplitude

[verbal sparrow]

Well normally amplitude is the distance from the middle to the top/bottom of a sin or cos graph. Basically, it describes how stretched it is vertically. But tan doesnt have a top or bottom cuz it goes infinitely, but you coukd describe how stretched vertically it is (which would be like multiplying by some number). In this case, the graph isnt stretched vertically, so its like multiplying by 1

<@&268886789983436800>

[rough wasp]

why did u call mods

[verbal sparrow]

I clicked on one of those by accident, and I think i heard of people getting a virus just from clicking an image so I got scared lol

Another scam

[rough wasp]

virus from clicking an image?

[verbal sparrow]

I could be mistaken

[pastel garden]

i think he means from clicking an attachment link?

[verbal sparrow]

Maybe an image link

[rough wasp]

i hope those scammers rot

[verbal sparrow]

Like to a website

[rough wasp]

ok i get the multiplying by 1 thing, alright well thank you for your help so much

[verbal sparrow]

[tawdry kraken]

!done

[midnight plank]

If you are done with this channel, please mark your problem as solved by typing .close

[rough wasp]

.close

[midnight plank]

Channel closed

Closed by @rough wasp

Use .reopen if this was a mistake.

[rough wasp]

i wanted to say bye to weaboo

[tawdry kraken]

Dang weeaboo was one mobile

Writing all that from mobile must have been killer

[midnight plank]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[ivory peak]

i have a confusion in this....will it be complete like 0 is a limit point and the cauchy sequence does converge to 0 but in whole space 0 doesn't even exist...and it's not like the whole space is subset of a bigger one too so...and like whole metric space is closed too I'm not really getting it

[visual tiger]

Wdym by "0 is a limit point"?

If a sequence in X is cauchy (with respect to d), then it would have to converge in X

That is if X is complete

[ivory peak]

i mean if I tend x to inf then it would converge to 0

[twilit jetty]

and what would y be

[ivory peak]

any other term

[visual tiger]

There's some idea in this, but we need a clear sequence

So x_n = ...

And then compute |x_n - x_m|

[twilit jetty]

already youre thinking of a sequence if youre saying "tend x to inf"

you just gotta mention an example of such a sequence

[ivory peak]

i mean shouldn't I look for a case where it would kind of disprove completion

[twilit jetty]

we're doing that arent we

[visual tiger]

That's literally what we're doing

[ivory peak]

yes

[visual tiger]

Finding a cauchy sequence

That doesn't converge

[twilit jetty]

for the record, do you personally believe (X, d) is complete (mb)?

[visual tiger]

Well... give us such a sequence

[visual tiger]

Did you omit "complete"?

[ivory peak]

yes it is it follows triangle ineq

if it had 0 included then ig yes

[twilit jetty]

including 0 isnt exactly clear rn so we're going to leave the exact 'form' of 0 out

[visual tiger]

You think adding 0 to X would make it complete?

[twilit jetty]

you must mean a different kind of 0

something to do with that 1/x and 1/y maybe

[ivory peak]

yes ig

[twilit jetty]

if you dont have an exact idea of what form this 0 would take, then we can find out what it exactly looks like later on

at least, you dont think this "zero" would be in X

so you dont believe (X, d) is complete?

[ivory peak]

yes it is cauchy as xn.xm would tend to increase but idk about convergent

[twilit jetty]

(X, d) is incomplete, so we'll have to look for a counterexample Cauchy sequence that can show that

[ivory peak]

okay

[twilit jetty]

lets try something first

[ivory peak]

okk

[twilit jetty]

you said to have x -> infinity

name me a sequence that -> infinity then

[ivory peak]

yes

[twilit jetty]

it can be really simple

[ivory peak]

identity fn

[twilit jetty]

thats not a sequence, thats a function

whats the domain?

[ivory peak]

[1,inf)

[twilit jetty]

f(x) = x is not a sequence though

for one, youre allowing for any real number index

since you said [1, inf) is the domain

[ivory peak]

i meant in a sense taking all values as it is

[twilit jetty]

for example, I can say the 4.45th term is 4.45

[ivory peak]

yes in that way

[twilit jetty]

just list the terms of sequence

[ivory peak]

okay

[twilit jetty]

sequences usually only have 1st terms, 2nd terms, 3rd terms, 4th terms, etc.

your sequence currently has an uncountable number of terms

you should just settle with the sequence 1 2 3 4 5 6 7 ...

[ivory peak]

okayy

[twilit jetty]

now before we continue,

[twilit jetty]

heres how you would say this sequence your way

we have $f(x)=x$ as before but we have the domain be the positive integers

[grand pond]

mtt

[twilit jetty]

{1, 2, 3, 4, 5, 6, 7, ...}

we read this function as f(index) = term at that index

so f(7) is the 7th term (which is 7)

another example of a sequence is f(x) = x^2 with the domain of positive integers

[ivory peak]

okay

[twilit jetty]

this would represent the sequence 1, 4, 9, 16, 25, 36, 49, ...

the 8th term is f(8) = 64

make sense?

[ivory peak]

yesyes

[twilit jetty]

alr be sure to choose the correct domain next time

alr we've got our identity sequence 1, 2, 3, 4, 5, ...

now lets match this with the usual notation:

[ivory peak]

okk

[twilit jetty]

Define the sequence ${x_n}_{n=1,2,...}$ as $x_n=n$

[grand pond]

mtt

[ivory peak]

okk

[twilit jetty]

now you were mentioning something about Cauchy about this sequence

[ivory peak]

yes

[twilit jetty]

especially with stuff like this

so what do you want to prove about xn

[ivory peak]

that after some N dist between xn and xm will be less than epsilon

n,m being greater than N

[twilit jetty]

well you technically never said you wanted to prove that xn is Cauchy

you gotta mention that

[ivory peak]

okok

[twilit jetty]

but yea we gotta show xn is Cauchy

[ivory peak]

ok

[twilit jetty]

alr go try this out

remember your distance is d(x, y) = |1/x - 1/y| instead of |x - y|

[ivory peak]

yess

[twilit jetty]

you should be able to figure out an N

[ivory peak]

N which is really large

[twilit jetty]

it doesnt have to be large

it just has to be good enough based on your epsilon

[ivory peak]

okk

in what way do I have to figure out

[twilit jetty]

have you done epsilon-delta proofs before

[ivory peak]

yes

[twilit jetty]

you always had your delta be some function of mainly epsilon, right

[ivory peak]

yes

[twilit jetty]

now this isnt really showing that a sequence converges, just that its Cauchy

so instead of "epsilon-delta" we have "epsilon-N"

that only changes the delta to an N really

so in place of a delta, we have xm and xn with m, n > N

now as before, you should try and find an expression for N based on epsilon

where when m, n > N, its always the case that d(xm, xn) < epsilon

[ivory peak]

N= n2 . epsilon?

[twilit jetty]

wdym by n2

@ivory peak hello?

[ivory peak]

kind of lost touch with epsilon delta proofs

[twilit jetty]

n2?

[ivory peak]

n is any term

[twilit jetty]

usually when you write n2, that means $2n$

[grand pond]

mtt

[ivory peak]

and n^2

[twilit jetty]

it means n^2 if you never learned the ^ symbol ebfore

youre not at that age anymore

you need to use ^ for power

[ivory peak]

okay sorry

[twilit jetty]

go write it again with the ^

[ivory peak]

N= n^2. epsilon

[twilit jetty]

alr now go show your work with this n^2 . epsilon

@ivory peak you here?

[ivory peak]

yess

[twilit jetty]

the n^2 . epsilon doesnt exactly make sense

since you have to start with an epsilon then pick an N

[ivory peak]

oh

[twilit jetty]

think about it like if you have |x - a| < delta,

you wouldnt know what the x is

same thing for the m and n

that being said youre a bit close

lets take an example

lets say epsilon = 1/2

now the requirements of a Cauchy sequence are:

,,\forall\epsilon>0,\exists N\in\mathbb N,\forall m,n\ge N,d(x_m,x_n)<\epsilon

[grand pond]

mtt

[twilit jetty]

compare that to the epsilon-delta version:

,,\forall\epsilon>0,\exists\delta>0,\forall|x-a|<\delta,|f(x)-f(a)|<\epsilon

[grand pond]

mtt

[ivory peak]

hmm yes

[twilit jetty]

(this isnt exactly how you write it but I made them look similar)

[ivory peak]

okk

[twilit jetty]

now as before, you begin with epsilon and a

and then you figure out a delta based on that epsilon and a, right?

[ivory peak]

yes

[twilit jetty]

this time you just have epsilon

that being said you did square the n

[ivory peak]

yes

[twilit jetty]

where did the squaring idea come from? it could work

[ivory peak]

because both terms are greater than N so took x= n and and other would be n+N and then ig I just assumed N to be small so I ignored it

[twilit jetty]

thats not really a proof

lets try out epsilon = 1/2 now

[ivory peak]

okk

[twilit jetty]

I dont know if youve done this before, but maybe youve heard of epsilon-delta as a game

I say epsilon = 1/2, you say a (positive) delta where |x-a| < delta means |f(x) - f(a)| < epsilon

[ivory peak]

okk

[twilit jetty]

now instead of that, we're doing this for a Cauchy sequence

[ivory peak]

okk

[twilit jetty]

so I say epsilon = 1/2, and you then need to figure out an N where m, n > N would mean d(xm, xn) < 1/2

now as before, you should try simplifying some things before trying this out

what does d(xm, xn) < epsilon simplify to?

[ivory peak]

(2.m)/(2+m) \leq n

[twilit jetty]

you need to mention the other requirement

[ivory peak]

for epsilon = 1/2

[twilit jetty]

2m / (2 + m) < n isnt the only equation you were supposed to find

go show your work on how you got 2m / (2 + m) < n

[ivory peak]

n \in ( 2/m(m+2), 2/m(2-m))?

[twilit jetty]

go show your work

also thats not what youre intended to find

youre skipping quite a lot

[twilit jetty]

this would not be useful to find an N

you should show more work to leave more of it useful to get an N from, ok?

you should be showing all of your work actually

[ivory peak]

ok

[twilit jetty]

so this time go show your work

[twilit jetty]

how did you get this

I know youre not going to show a screenshot, so Im going to ask you to explicitly show the steps you write down

instead of just "yeah uhhh heres what I was doing"

[ivory peak]

opening the d(xm,xn) \leq ep

[twilit jetty]

ok thats not useful

for one, I opened it up and I got an N

you didnt

[ivory peak]

okk

[twilit jetty]

instead of just "yeah uhhh heres what I was doing"

Im going to ask you to explicitly show the steps you write down

[ivory peak]

okk

[twilit jetty]

picture works if you wrote it down

it should be really quick to send your work over, like 15 seconds quick

its just wherever you wrote stuff down

[twilit jetty]

see that wasnt that hard

you need to be doing this all the time next time

[ivory peak]

okk

[twilit jetty]

alr remember what youre trying to find

[ivory peak]

N?

[twilit jetty]

you need to find an N where m, n > N would always guarantee that |1/m - 1/n| < 1/2

[ivory peak]

yes

[twilit jetty]

now what you wrote down isnt exactly useful for that purpose

for one, this is negative

[ivory peak]

oh yeah

[twilit jetty]

we'll need a different idea if we want to get this to work

[ivory peak]

okok

[twilit jetty]

heres a different idea you can try out

what does |this - that| represent?

[ivory peak]

nbd of this?

[twilit jetty]

??????

you know I cant read your mind

[ivory peak]

sorry I'm used to using neighbourhood as nbd

[twilit jetty]

then thats not correct either

[ivory peak]

ohk

[twilit jetty]

tell me what a neighborhood would be then

[ivory peak]

wait i thought in other way it probably means the distance

between this and that

[twilit jetty]

yea thats all it means

neighborhood is unnecessary

[ivory peak]

yes

[twilit jetty]

the definition of neighborhood is instead |this - x| < r

r is fixed, and the x where that is true would be in the neighborhood

[ivory peak]

okok

[twilit jetty]

I didnt even say < r

so I cant be talking about a neighborhood ok?

[ivory peak]

okkk

[twilit jetty]

so |this - that| represents how far apart this and that are

[ivory peak]

yess

[twilit jetty]

now the this and the that are 1/m and 1/n

now lets think, what would be a lower and upper bound of 1/m and 1/n?

[ivory peak]

lower 0 and upper 2

noo wait

1 upper

[twilit jetty]

alr, so 1/m and 1/n are always numbers inside (0, 1], right

[ivory peak]

yess

[twilit jetty]

now if we require that |1/m - 1/n| < 1/2

that would mean the 1/m and the 1/n are less than 1/2 apart

[ivory peak]

yess

[twilit jetty]

now do you think that would be possible if 1/m and 1/n were in (0, 1]?

[ivory peak]

yes if I take both n,m greater than 2

[twilit jetty]

just > 2 is ok

[ivory peak]

ok

[twilit jetty]

look at that, you found an N

[ivory peak]

oh

[twilit jetty]

the N (which is 2) is based on the epsilon (which is 1/2)

now where did you get that 2 from?

[ivory peak]

okay

[twilit jetty]

just keep elaborating until I think youre good

[ivory peak]

since the distance has to be less than 1/2 if I take m large enough and keep n constant at max i.e 2 since it has to be less than 1/2

then after then I generalized n too

since the absolute diff will be stay less than 1/2

[twilit jetty]

that would require some formula for m and n together

whats the exact one you used

[twilit jetty]

for eaxmple, where did you get the "max i.e. 2" part from

[ivory peak]

i just thought that if n is 1 (since n is natural) it would not work for every m...so to generalize n=2 works...so triangle inequality ig

[twilit jetty]

thats good enough

if you had m or n = 1

then |1 - 1/3| > 1/2

however if you have m, n > 2,

[ivory peak]

yes

[twilit jetty]

what would a new upper bound of 1/m, 1/n be?

[ivory peak]

1/2

[twilit jetty]

and since 1/m and 1/n are always in (0, 1/2],

that assures us that |1/m - 1/n| < 1/2

right

[ivory peak]

yes

[twilit jetty]

now we've got something good

lets try this for any epsilon

first, lets say epsilon is always 1 / natural number

what would you assign N to be?

and remember for epsilon = 1/2 you came up with N = 2

[ivory peak]

yes

N = 1/epsilon

[twilit jetty]

yep

so lets say m, n > 1/epsilon

then 1/m and 1/n would have an upper bound of?

[ivory peak]

epsilon

[twilit jetty]

and from before, 0 < 1/m and 0 < 1/n

so since 0 < 1/m < epsilon and 0 < 1/n < epsilon,

[ivory peak]

yess

[twilit jetty]

we would expect to then figure out that |1/m - 1/n| < epsilon

right?

[ivory peak]

yes

[twilit jetty]

now that would be the whole proof, so it remains for us to prove that

essentially you need to prove that:

x, y in (0, epsilon] => |x - y| < epsilon

this is more of algebra than anything real

if you want, you can do (0, epsilon) instead since it seems you were taught m, n > N instead of m, n > N

[ivory peak]

okl

thank you I will look more into it @twilit jetty sorry for the trouble

.close

[midnight plank]

Channel closed

Closed by @ivory peak

Use .reopen if this was a mistake.

[twilit jetty]

np, but I was just waiting for you to tell me whether you figured out the proof or not @ivory peak

.reopen just in case

[midnight plank]

✅ Original question: #help-49 message

[ivory peak]

I did figure it out but I just realized I need to work more on my basics and tomorrow I have an exam so kind of time bounded...but thank you again

[twilit jetty]

ok thats good

were still sort of not done because of some details about this proof

the major one is that this is not at all a proof that you would expect at first

[ivory peak]

i will ask within a week if it would be okay

ik

[twilit jetty]

you can ask another question any time, but for now lets focus on how this proof went

you know the way you proved this is sort of familiar to how I used to do things

in high school you usually learn a list of problems and a list of methods

then you just connect the dots, method to method, until any problem can be broken apart into methods that solve them

usually the problem makes this clear, you get more used to what a method can do and how it can be used as you move on

does this make sense so far?

[ivory peak]

yes it did

[twilit jetty]

now youre presented with a proof, which unfortunately is a lot more open-ended

for a proof, you also need to allow intuitive ways to do things in as well

[ivory peak]

hm yes

[twilit jetty]

for example theres no way the algebra way of solving this (which got you those two bounds on n) wouldve figured this out

we had to take a different perspective into things

thats also why I figured what you mentioned at first could be useful

[ivory peak]

yes

[twilit jetty]

you mentioned theres a "zero" that was supposed to be in X but isnt?

[ivory peak]

ig I will reopen .reopen

[twilit jetty]

it was already opened

[ivory peak]

oh

[twilit jetty]

I did .reopen earlier

[ivory peak]

thanks

[twilit jetty]

I want you to try and take a guess at what this 0 should be

heres a hint

X is [1, infinity)

but the distance does 1/ the number before doing |this - that| on them

its |1/x - 1/y| after all

[ivory peak]

yes

[twilit jetty]

so what would you include in X here?

it may not be a number

think about where the numbers of X become after you 1/ them

[ivory peak]

i what sense

[twilit jetty]

theres no other sense than including an extra element in X to make (X, d) complete

that element isnt a real number

what would it be?

[ivory peak]

so we need to include an element which isn't real ?

[twilit jetty]

its not complex either

its not considered a number

remember from earlier?

whats the definition of complete?

[ivory peak]

every cauchy seqn in X is convergent in X

[twilit jetty]

alr, and 1, 2, 3, 4, ... is a Cauchy sequence, right?

[ivory peak]

yeah in metric d

given

[twilit jetty]

what does 1, 2, 3, 4, ... converge to?

[ivory peak]

it doesn't I meant in 1/n sense

[twilit jetty]

what does

1, 2, 3, 4, ...

converge to?

we know its not in X, we've proven that

[ivory peak]

it doesn't converge

[twilit jetty]

it doesnt look like youre opening your mind enough to new ideas

[ivory peak]

wait wait

[twilit jetty]

its not in X, but X is every number 1 and above

we know it doesnt converge to a number of any sort

if only we had a symbol for it

1, 2, 3, 4, ... is said to diverge if we cannot find something it converges to

[ivory peak]

yes

[twilit jetty]

but lets say we include things that arent numbers

then 1, 2, 3, 4, ... will converge

what does 1, 2, 3, 4, ... converge to?

[ivory peak]

inf?

[twilit jetty]

yes

[ivory peak]

okk

[twilit jetty]

we can use the extended reals

([1, infinity], d) would be complete instead

the easiest way to notice this is:

d always does 1/ the numbers before measuring the distance the normal way

so instead of X, we can consider 1/X, the version of X that d only seems to use

1 / [1, infinity) is (0, 1], isnt it

[ivory peak]

yess

[twilit jetty]

and (0, 1] isnt even closed

so it cant be complete

for example, 1/2, 1/3, 1/4, ... -> 0

and 0 isnt in (0, 1], right?

[ivory peak]

yes

[twilit jetty]

now we consider the X version

we have 12, 3, 4, ... -> infinity

and infinity isnt in [1, infinity)

if we were to include infinity in the set,

then 1 / [1, +infinity] = [0, 1]

and that way we can be certain itll be complete

[ivory peak]

yes

[twilit jetty]

keep in mind this defines 1/0 = infinity and 1/infinity = 0

[ivory peak]

ok

[twilit jetty]

this isnt always the case you know

so far we stuck to only working with positive numbers for our problem

if the numbers were also negative, 1/-infinity = 0 too and we'd be in a bit of a problem with 1/0

[ivory peak]

yesd

[twilit jetty]

so we're only using infinity in the extended reals to represent an element that would be the limit of 1, 2, 3, 4, 5, 6, ... and any other otherwise diverging sequence

[ivory peak]

yes

[twilit jetty]

now naturally youd have to confirm again that this new (X u {infinity}, d) is still a metric space

to be honest, it might not even be a metric space anymore

[ivory peak]

okay

[twilit jetty]

now regardless of whether we consider infinity or not,

the fact that 1 / [1, infinity) = (0, 1] which is not closed is important

that alone is what showed me where to go in this problem

the distance function always does 1/

and in the 1/ version, we end up with numbers that arent in a closed set

1/m and 1/n can get as close to 0 as they like but never 0 itself

[ivory peak]

yes

[twilit jetty]

noticing a thing like this is not easy, and I dont think it can be taught

you notice more convenient patterns or even more abstract promising details over time

the usual algebra ways wouldnt have shown any of this

[ivory peak]

hmm okay

[twilit jetty]

now finding those facts on the spot may not be easy, but you could settle with the next best thing

you consider them as tricks, then you remember them as extra methods you can use

you add to your 'dictionary' of methods you try out

[ivory peak]

okok

[twilit jetty]

each new trick you see, you try and use and see the point of doing

then that way, now you can use the same trick just as they used it

sometimes the problems in your homework/test will be structured this way:

1(a): identity or fact that seems a bit general
(b): harder problem that uses 1(a) as a trick

its trying to get you to recognize, then use, something useful

[ivory peak]

okay okay

where I am confused is

that how should I even imagine space like X

that

the result distance function gives

should be in space or not

[twilit jetty]

the result distance function gives is a distance

X is just where the points are

X does not include how far apart they are

that would be d(X, X)

for example, d(1, 2) = 1/2

1/2 isnt in X, doesnt have to be

1/2 is in d(X, X), the set of all distances you get out of X

d(X, X) doesnt need to have anything to do with X

[ivory peak]

ohh

ok

[twilit jetty]

I shouldve told you that much earlier

I forgot

[ivory peak]

so the reason it's not complete is because it leads to inf so 1/inf leads to 0 and 1/x can be never be 0?

[twilit jetty]

you can view it that way

you can also view it like "the distance function is the same as |x - y| for 1/X, and theres no way |x - y| for (0, 1] is complete"

[ivory peak]

ohhk

[twilit jetty]

either way works

youre essentially saying that 0 shouldve been in d(X, X) for that sequence but it wasnt

0 is in d(X, X) though

you can do d(2, 2) = 0

[ivory peak]

so what if I take another metric space and the sequence converges to a certain number that is not in space...would it still be complete...( not talking about dist )

[twilit jetty]

by definition it would not be complete

complete spaces mean any cauchy sequence will always converge to something in the space

you find a cauchy sequence that converges to something outside the space, its not complete

[ivory peak]

like whole space is rationals only no reals subset or anything then I form a seqn converging to root 2

[twilit jetty]

yep

[ivory peak]

so basically if in a metric space I'm left hanging it's not complete

[twilit jetty]

yea

just as before, you were looking for hanging things

1, 2, 3, 4, ... is Cauchy because the distances -> 0
but what ic vonerges to (infinity or no number at all) isnt in X

[ivory peak]

okok

thank youu

I got it esp dist part

[twilit jetty]

nice

and you look for ways through instead of just methods

[ivory peak]

okok

thank you again @twilit jetty

. close

[twilit jetty]

np

.close

[midnight plank]

Channel closed

Closed by @twilit jetty

Use .reopen if this was a mistake.

[twilit jetty]

dont leave a space between the . and the close

[midnight plank]

Available help channel!

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[twilit field]

Quick question in part (2). That's just Z mod 2 right

$Z /2Z = {a + 2\Z \mid a \in \Z }$

[grand pond]

wai

[runic hamlet]

yes

[twilit field]

I suppose instead of multiplying it first and then modding each term by 2 I can do it beforehand?

[olive matrix]

yes

[twilit field]

so p(x) mod 2 is really x^2+1

as -3mod 2 is 1 and -5 mod 2 is 1

and 7x^3+33x-4 is x^3+x

[runic hamlet]

if you have multiplied it out in (a) already then just reduce afterwards...

[twilit field]

I didn't do (a) as that's just regular poly multiplication

[wind oxide]

1 help channel is crazy

[twilit field]

Hey, I'll close this now

Thanks a lot everyone!

.close

[midnight plank]

Channel closed

Closed by @twilit field

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[hard shard]

why is 92 half of 99?

[grand gust]

why are you becoming the very thing you swore to destroy

[bold peak]

<@&268886789983436800> troll

[hard shard]

these guys were playing the game run escape, im so confused

[rancid wolf]

1+1=2

[hard shard]

nobodys here, so im enjoying the empty space

[bold peak]

Actually does an edit ping ping the mods still

I'm not sure

[hard shard]

no

[bold peak]

Cool

[rancid wolf]

Yes ofc

[bold peak]

I'm pretty sure it didn't but wanted confirmation

[lilac finch]

ping again

lol

[rancid wolf]

It does

[hard shard]

@shadow scaffold lying

[rancid wolf]

Wth

[tawdry laurel]

[rancid wolf]

Rude

[tawdry laurel]

Oh thats so cute

[bold peak]

[tawdry laurel]

[bold peak]

Yes I've asked for that one too

And

[lilac finch]

[bold peak]