#Proof of sumpremum real analysis
atomic scrollBOT
fallow patrol
What does it mean for Y to be bounded above? Can you find a bound?
frozen glade
@fallow patrol my only thought is, that it must be unbounded? if x must be greator than 0, it can still be 0.0000.... , which put at the bottom of that fraction would give +infinty, so i dont understand how it could be boundd
fallow patrol
But you know that inf X > 0
frozen glade
@fallow patrol obiously intueitivley i can see that its asking me to show that the supY is 1 right?
fallow patrol
Sup Y is not 1
frozen glade
it wouldnt be the recipricol of inf X 1/0?
ah because its not 0 but it is greater than 0
idk
fallow patrol
Yes, it is the reciprocal of (inf X).
frozen glade
so if X is greater than 0, and x is in X, that means that 1/x could be infinitly small or large? because 1/v small x = infity , 1/very big x = -infity
that is why im confused as to how there could be an upper bound
fallow patrol
so if X is greater than 0
X isn't an umber
frozen glade
right but in the Set Y it states that x must be greater than 0 correct?
1/x where x must be greater than 0
because x is in X
frozen glade
so how would i think of an upper bound
fallow patrol
The question actually gives you an upper bound
frozen glade
does inf(X) ^-1, suggest x<0?
fallow patrol
no, but you don't need to find it. It's the infimum of X.
frozen glade
ok hmmm
thanks for helping btw
1/x , where x is bigger than 0
im still at the problem where if x is less than 1, then it could be infinitly large?
idk am i thinking of this wrong?
fallow patrol
it's true that 1/(inf X) could be arbitrarily large, but it can't be infinitely large
frozen glade
right
fallow patrol
(inf X) is a number greater than 0. So 1/(inf X) is just some number. It doesn't matter what that number is
frozen glade
so your saying that any upper vound would work?
larger than 0
any number larger than 0 is an upper bound?
hence 0 is the supremum?
fallow patrol
Pretend that X is the interval (1/100, infinity)
frozen glade
ok
fallow patrol
and do that question first, then use the same strategy but with variables for the original question
frozen glade
so would u define the upper bound of Y, not as a number but expression involving the set X?
ok so
so we have to prove infY= 1/x+inf X?
or something like that
Prove sup Y = 1/sup X
and tha 1/Inf X is an upper bound for Y
and that anynumber < 1/inf X is not an upper bound for Y?
am i on the right lines idk
frozen glade
@slate hull ok, why is inf X its upper bound
slate hull
<@1000514315113218049> ok, why is inf X its upper bound
Inf X is an upper bound for Y because it is greater than or equal to every element of Y
frozen glade
ah i see, because every term of Y would be 1/x
slate hull
boom
slate hull
lets see
y would be bigger
than 1
this is because the denominator of the expression y/(y-√x^2) is smaller when x is smaller Therefore, as x gets smaller, the expression gets bigger.
therefore, if x was 0.05 in the equation y/(y-√x^2), then y would be bigger than 1.
frozen glade
where did u get that equation from
maybe i need to look up on this more myself, what topic would this comparing infimums and suprmums come over
ive googled but couldnt find
slate hull
where did u get that equation from
myself ive figured
slate hull
maybe i need to look up on this more myself, what topic would this comparing inf...
order theory
frozen glade
thanks
slate hull
yw
slate hull
let me see
it shows that the infimum of the set Y is less than or equal to the infimum of the set x so its correct
frozen glade
ok thanks
frozen glade
where u live
slate hull
major?
frozen glade
idk we say major
idk what that is
BsC
Bachollers
and the one above you just do a year extra
Msc masters
frozen glade
idk bro
lol
can do alot with a maths degree
will probs just try get some remote job and live somewhere cheap$
slate hull
im going to chemical engineering and getting a phd and taking over my dad gas miners
slate hull
i lowkey just joined this server
frozen glade
ye i seen lol
frozen glade
lol , im doing this work ive procrastinated will probs b up all night
u wanna get ur dad to get me a scholar ship in abu dhabi and ill come work for him aha
frozen glade
forigners tho?
slate hull
like no intrest
fallow patrol
it shows that the infimum of the set Y is less than or equal to the infimum of t...
just fyi, this doesn't make any sense
it's not even necessarily true
frozen glade
O dog was teahing me well tbf
fallow patrol
this argument is still correct, if a bit hard to follow
frozen glade
O dog, is the thing i posted correct, im going to go learn about order theroy
and use it as an example
ah ok
slate hull
if X = (.1, 1)
y/(y-√x^2) = 1.6
2.667√x^2 / (2.667√x^2 - √x^2) = 1.6
2.667√x^2 / 1.667√x^2 = 1.6
1.6 = 1.6
so, the answer y=2.667√x^2 satisfies the original equation.
fallow patrol
Wait, is that chatgpt? It looked good at the start, but this line is insane
frozen glade
it is ai yeah
do u mind checking something i did myslef ? to see if im right with regular infimum stuff?
fallow patrol
don't use ai on things you don't already understand very well
slate hull
wait whats that
fallow patrol
y/(y-√x^2)
Where does this expression come from?
slate hull
> y/(y-√x^2)
Where does this expression come from?
comes from the original equation (y - √x^2)* = 1.6
by dividing both sides by (y-√x^2)
y - √x^2 = 1.6
If we divide both sides of this equation by (y-√x^2), we get
(y - √x^2)/(y-√x^2) = 1.6/(y-√x^2)
fallow patrol
But the question was this:
frozen glade
i know but i am now doubting myself in proving suprmemum lol
if what i did was right i will move onto studdying this order theory stuff
if not i will have to revisit
slate hull
But the question was this:
he was talking abt something else asking me if it was correct\
frozen glade
oh i see :/
slate hull
@fallow patrol
fallow patrol
this is all backwards
logic doesn't work like this
If you want to show something, you can't start by assuming it's true
It's good to know what expression you're trying to prove, but that's not the proof
frozen glade
its not?
fallow patrol
If Shakespeare was eaten by aliens, then Shakespeare would be dead.
Shakespeare is dead.
Therefore Shakespeare was eaten by aliens.
frozen glade
the question told me its the upper bound tho
i see what ur saying, but how else would i prove this, everything online tells me to do like this
fallow patrol
I mean the equations
frozen glade
but how would i ever know to stat from 1/4 >=0
fallow patrol
because what you have here is scratch work that found out that you can start with 1/4 >= 0
frozen glade
ah ok, yeah i did scratch work for the second bit
and then started the proof
in the other images im talking about
are those parts right?
fallow patrol
Your proof should not include that scratch work
frozen glade
oh ok
fallow patrol
but yes, this part is much better
frozen glade
ok so if i get rid of the bit above that i shouldnt drop marks?
all that matters is the proof?
and ill rewrite the top bit starting from 1/4
fallow patrol
the scratch work is how you find the proof, so it's important
but it's not part of the proof itself
frozen glade
i see
was ama correct that the previous question we were talking about was the topic called order theory?
fallow patrol
Kind of, but not in a useful way.
Order theory is studying binary relations such as "is less than"
frozen glade
what would u recommend searching to find what we were looking at
fallow patrol
But order theory is much more general, and you don't need to learn anything from order theory that wouldn't already be considered basic algebra
frozen glade
cuz i was looking and couldnt find is why i came here
fallow patrol
If a > b, then what can you say about -a and -b?
frozen glade
-a>-b
fallow patrol
You could call this a question about order theory, but I think that's massive overkill. This is just middle/high school algebra
frozen glade
gotcha
-a<-b
i dont mean to pester you, would but would u mind checking some of my other work to see if its correct, no worries if not :)
fallow patrol
sure, I'll look at another
frozen glade
This is an induction
fallow patrol
I don't see what your agument here is
remember that you always want to end on the thing you're trying to prove
so you probably want to start with something like 1-x^2 < 1
and then note that dividing by 1+x does not flip the signs since 1+x > 0
the language here doesn't make sense
You want to assume that the inequality is true for some n = k
frozen glade
is this better or are you saying to fully re -write back to front
fallow patrol
rewrite it in the opposite order
your implications are going in a direction that is irrelevant
frozen glade
i see ok
fallow patrol
It's the same reason that this argument below is nonsense:
If Shakespeare was eaten by aliens, then Shakespeare would be dead.
Shakespeare is dead.
Therefore Shakespeare was eaten by aliens.
frozen glade
so i should re-write the indectuive step section aswell?
fallow patrol
Your inductive step is actually okay because the thing you're starting with is the inductive hypothesis
You have tons of extra scratch work that should be removed, though
frozen glade
as in like, too many little steps that are obvious?
fallow patrol
not exactly
the parts where you did calculations that lead you to know what to put in your proof
frozen glade
ok i got u
fallow patrol
a lot of the "we will prove this to show that"
this reads more like an explanation of how to prove it than an actual proof, if you know what I mean
frozen glade
yeah got u
slate hull
dam o dog you crazy
fallow patrol
I run into this problem as a teacher, because when you're explaining it, you include all those extra details because the students need to know it. But then the students don't need to repeat all those extra details because they're not explaining the problem, they're just doing it
frozen glade
what country u teach in bro
slate hull
what country u teach in bro
i bet you china or some
frozen glade
america or england im guessing, if not your english is really impressive
fallow patrol
I've taught a bunch of high school and university
My English is good because it's pretty much my only language
frozen glade
O dog i was wondering if you could help me on this last bit im stuck with
is proof of a teloscopic series
fallow patrol
post it
frozen glade
i used partial fraction decompoisition to get to that final series
then
i understand that these terms are supposed to cancel some how
e.g this is n=1, n=2, n=3
cant for the life of me figure our where
i know it converges to 3/2
fallow patrol
Not every terms cancels
because you'll have some left over that will give you the answer
look at all the terms with denominator 3
frozen glade
yeah
fallow patrol
then look at all the terms with denominator 4
frozen glade
-1/3 + 5/3 -4/3
fallow patrol
=0
frozen glade
ah
for a second i thought u put a shocked face
ahhaha
anyways @fallow patrol tysm for your help im going to go look up on that infimum shit. Have a good rest of your night!
fallow patrol
good night
frozen glade
.solved
atomic scrollBOT
Solved
frozen glade
@fallow patrol i did that question from eariler, does this check out?
fallow patrol
You don't need to say this part. That's something the teacher tells the student who is just learning this.
frozen glade
ok
