#sequence problem

384 messages · Page 1 of 1 (latest)

exotic sparrow
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I want to understand this question and relevant topic

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serene ospreyBOT
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sturdy warren
exotic sparrow
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No no. I want to understand all the options

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And the topic name related to it

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I want to learn the method actually

sturdy warren
exotic sparrow
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I only know about convergent and divergent

sturdy warren
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well you can look up subsequence, bounded, and cauchy sequence

exotic sparrow
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Now done

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I don't know about the abcd statement and their special examples

sturdy warren
exotic sparrow
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What is wrong in option A

iron tide
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if there is just one convergent subsequence that does not mean the initial sequence is bounded

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e.g 1 2 1 3 1 4 1 5 ...

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for ease of reading, write out the problem formulation in your question or take better pictures of it

exotic sparrow
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If you download the picture then it will be more clear if we have problems with pictures

slender magnet
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what

exotic sparrow
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I have studied sequences and solved some of their examples/questions

slender magnet
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did you read the books I've recommended

slender magnet
exotic sparrow
slender magnet
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or subsequences

slender magnet
exotic sparrow
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Yes. I read cauchy sequence

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Yes. I started reading the book which you suggested

slender magnet
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then this question really has no place for confusion

exotic sparrow
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Notes me padha hu bhai cauchy k nare me

slender magnet
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do you know what a subsequence is

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well let's look at 1

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"every sequence that has a convergent subsequence is bounded"

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if we had "sequence" instead of "subsequence" it'd be true

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(prove it)

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but as it's a subsequence! the rest of the sequence can go fuck itself

exotic sparrow
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Bhai hindi m likha do

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Mujhe english kam samjh me aata

real onyxBOT
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Coffeλ

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Coffeλ

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Coffeλ

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slender magnet
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ok

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aab aya 2

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2 ka sidha sidha dekhlo

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$a_n = n\cos(1/n)$

real onyxBOT
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Coffeλ

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slender magnet
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cos(1/n) ke baare me kia bol sakte?

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cos(x) 1 se 0 ke beech me kaisa behave karta?

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,w graph cos(x)

real onyxBOT
exotic sparrow
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Oscillation karega?

slender magnet
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increasing strictly between 1 and zero (zero to 1)

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ulta bola

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1 to zero

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toh cos(1/n), 1 ke aas paas ayega

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but n toh bas baare jaa raha he na?

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is liye sequence hi monotone increasing he.

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4 thoda tricky he

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but it's exactly similar to 1

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it has a cauchy subsequence i.e. the convergent subsequence!

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because convergence => cauchy :3

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but the whole sequence might not be cauchy

slender magnet
exotic sparrow
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Bhai ye example kha se aaya apko dimag me?

slender magnet
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ek sequence ke baare me socha jisme se ek part, ek index baad karke zero hojata

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@exotic sparrow aur bhi ek accha example de raha

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First we look at the set

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$\qty{\frac{1}{m}+\frac{1}{n} : m, n \in \mathbb{N}_{>0}}$

real onyxBOT
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Coffeλ

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slender magnet
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this is countable

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we want to make a sequence that covers every term here

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how would we do it

iron tide
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what's your experience with math @exotic sparrow your questions are so often about definitions and basic technique. are you an undergrad? high schooler?

slender magnet
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$a_{11},a_{1,2},a_{2,1},a_{1,3},a_{2,2},a_{3,1},\ldots$

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@iron tide do you know of a simple closed form expression for each term

real onyxBOT
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Coffeλ

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slender magnet
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it's just the usual way to sequencize! the Cartesian product of 2 countable sets

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$\qty{\frac{1}{m}+\frac{1}{n} : m, n \in \mathbb{N}_{>0}}$ is just like $\mathbb{N}\times \mathbb{N}$

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without the zeroes

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how do i say this without butchering notation

iron tide
real onyxBOT
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Coffeλ

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slender magnet
slender magnet
real onyxBOT
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Coffeλ

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slender magnet
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and guess what if we use the set $\qty{n+1/m : n, m \in \mathbb{N}_{>0}}$ and put it in a sequence, we get a diverging sequence with infinite converging subsequences!

real onyxBOT
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Coffeλ

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slender magnet
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$\qty{n+1/m:n,m\in \mathbb{N} _{>0}} = \bigcup {n\in \mathbb{N}\setminus \qty{0}}\qty{n+1/m : m\in \mathbb{N}{>0}}$

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@exotic sparrow you there?

exotic sparrow
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Yes. I am here

slender magnet
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oh no

real onyxBOT
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Coffeλ

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slender magnet
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so a sequence can have a lot of converging subsequences

iron tide
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$$ \sum _{i=0}^\infty \sum _{j=0}^\infty a(i,j) = \sum _{i=0}^\infty \sum _{j=0}^i a(i-j,j) $$

real onyxBOT
iron tide
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if that's what you meant

slender magnet
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I want a closed expression for each term

slender magnet
iron tide
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well ya it's counting elements of Q along the diagonals basically

slender magnet
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so I can show it to Arjun

iron tide
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not that I know of

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or at least it looks ugly

slender magnet
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i guess

iron tide
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it'll be a countable union of pieces of length 1, 2 , 3 etc

slender magnet
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mhm

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anyways

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I'm done lazing around for now

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@exotic sparrow you done?

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as in do you understand now

exotic sparrow
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I am still here

slender magnet
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samajh me aaya?

exotic sparrow
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Ha aa toh raha hain lekin ek bar me toh nahin mujhe time chahiye aur bar bar pdna pdega

slender magnet
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wdym

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he said "i can understand it but not in one reading, ill have to read it over and over over some time"

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@exotic sparrow why tho?

exotic sparrow
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For better understanding 😆

slender magnet
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dog what is better understanding to you

exotic sparrow
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I have a slow processor chip

iron tide
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fair enough, as long as you put effort into understanding solutions it's fine

slender magnet
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I'm asking you

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what is "better understanding" to you

exotic sparrow
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Matlab???

iron tide
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otherwise you will not learn and keep posting questions in the style of "how do I solve this" instead of "I have tried this this and that but it's not working"

exotic sparrow
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Jab tak me us cheez ko realise visuals aur feel na kar leta hoon

slender magnet
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"aur bhi acche samajhna" iska matlab kia he tumhare naate

exotic sparrow
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I didn't say that

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You took it wrong interpretation

iron tide
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remember that everything starts at definitions

exotic sparrow
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I said i will read ot twice and thrice so that i will understand it better

iron tide
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learn the definitions and relevant examples

slender magnet
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dog English is more important for you right now

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please learn English well otherwise you won't be able to read texts.

exotic sparrow
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Texts toh samaj aa rha hain par me express sahi nahin kar paata hoon

slender magnet
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despair yeah learn English

exotic sparrow
slender magnet
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halt math

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start an English course

exotic sparrow
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Still on my first priority 😍

slender magnet
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watch movies or something

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with captions

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will make you English master in a month

exotic sparrow
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I have learnt english so far through reading and talking with people on discord. I am learning a lot by discord. I am graduated in bachelor of science mathematics from government University where professor doesn't come a single day. I studied and got minimum marks so i have a degree but i haven't read proper text books and anything. But now i started again on my own.

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@iron tide

slender magnet
exotic sparrow
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Just went and passed papers

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In our university exams they just ask previous questions and via rote learning i git the degree 😭

slender magnet
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this is reassuring

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I can get a degree in math too

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yeee

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(hopefully)

exotic sparrow
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Bro you will top the university

slender magnet
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you have no idea

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about the 14 yr Olds

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they are crazy they will take over the world

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0 chance of my ass getting through

exotic sparrow
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Yesss yesss i was not aware until i came to discord

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14yr are superb 😍

slender magnet
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yes

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@iron tide what's your view on crazy 14yr olds

exotic sparrow
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Bro may I ask how old you are?

slender magnet
exotic sparrow
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You are solving lecture level questions actually with me

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🫣🫣

slender magnet
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gilbert it's really not something special, anyone can do it if they want to learn it

exotic sparrow
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You were right about "aage jake nahin phategi" moment

iron tide
exotic sparrow
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Yes yesss this dude was lying to me

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What about you? Al seems PG graduated to me 🫣

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A sequence can be made of two and more convergence subsequences

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But this is not possible in series i guess

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Because the general term will be destroyed m

slender magnet
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(in a bad way, I shouldn't have spondylitis)

exotic sparrow
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I have an interesting question because it is related to a series which contains these kinds of subsequences

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Check

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@slender magnet @iron tide

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Sr for ping

slender magnet
slender magnet
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but one way or the other

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it's related to the ratio test

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do you know what $\lim \sup$ is?

real onyxBOT
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Coffe𝑦

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slender magnet
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@exotic sparrow

exotic sparrow
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Yes. Ratio test

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Which shows us it is increasing to a limit or not

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When we divide two terms each other then we get a value if greater than one which means like series increasing or diverging so and when we get a value less than 1 inch the series is converging

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By mistake I wrote subsequences

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So I guess this question is wrong because two different types of sequences in a series is not possible

slender magnet
slender magnet
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wdym.

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do you know what the limit supremum is

exotic sparrow
slender magnet
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then what's sup

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$\lim \sup a_n \leq \sup a_n$

real onyxBOT
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Coffe𝑦

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slender magnet
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more importantly

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Bhai pura yehi samjhaate samjhaate time lagjayga

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yeh part kitab se karlo

exotic sparrow
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Ruko me likhta hoon

slender magnet
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tao wala sahi ho raha? agar woh aasan lage toh fir dusra pagro

exotic sparrow
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Aur supremo ka matlab yah hai ki koi bhi Jaise sequence hai to vah uska jo list upper bound hota hai aur lowest upper bond

slender magnet
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kiuki woh starting ke liye kitab he

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kuch part me kam dia

exotic sparrow
slender magnet
exotic sparrow
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Aur limit laga di sup toh

slender magnet
exotic sparrow
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Vo ek number dega

slender magnet
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hmm

exotic sparrow
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Agar finite hua toh convergence

slender magnet
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actually

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example deke batata

exotic sparrow
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1/n

slender magnet
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ya choro definition hi dedeta

exotic sparrow
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Vc aa sakte?

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Idhar kidhar

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If possible 🫣

real onyxBOT
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Coffe𝑦

390895986106564609.png
slender magnet
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@exotic sparrow

real onyxBOT
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Coffe𝑦

390895986106564609.png
slender magnet
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haa ab Sahi he

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ye cheez bohot kaam ki he

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ye sequence ke limit point ke bound he

exotic sparrow
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Acha

slender magnet
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toh diverging sequence ka sup toh infinity he

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but limit supremum na bhi ho sakta he!

exotic sparrow
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Agar limit exists kara toh sequence convergence?

slender magnet
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hmm bohot cheeze he yaha

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kitab se padho

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Kitna tak hua?

exotic sparrow
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Abhi toh sequence k example dekh rha hu book se 107 page par hoon

slender magnet
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btw

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ek kaam karna

exotic sparrow
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Sorry for taking your time 🫣

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M bhut jyada sawal puch liya padh rha nahin

slender magnet
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me yeh bata raha kiuki mujhe khud agar koi yeh batata toh accha hota

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  1. ek chapter padhke samajhne me time lagega, ye lim sup lim inf mujhe samajh ne me kam se lam 2-3 hafte lage bas 6 page! toh haar mat mano. Agar nahi samajh aye yaha puchna nahi toh time waste hoga.
  2. har exercise karo waha se aur proof jo exercise me diye he!
  3. proof writing sikhlo yaha se.
  4. proof writing haat me laane ke liye proof book se likho khud!
  5. har din kitna padha dm karke batana mujhe, me extra kuch bata dunga jo tao book me nahi he
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@exotic sparrow

exotic sparrow
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Thik hain ye acha tarika rahega mere liye...meri exam m vo log basic definition se jyada questions puchte hain...unka syllabus defined nahin bas b.sc m.sc kuch bhi puch sakte hain

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ताऊ की बुक अच्छी हैं

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Dono volume kam k hain vause

slender magnet
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volume 2 mat padho abhi

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vol 1 ke baad zakon karlo

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fir me vol 2 krunga

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kiuki vol2 me Fourier series, measure theory wager he

exotic sparrow
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Fourier series ka ratta mar k pass kiya tha 😭

slender magnet
exotic sparrow
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And vo laplase transformation

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Mujhe samjh toh sab aa jata haun lekin vhi haina khud se kuch ni kiya kuch

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Ungali pakdne ki aadth ho gyi

slender magnet
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hota he

exotic sparrow
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Bhai ye question me samjh gya

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Option B discarded

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Cos(1/n)/(1/n) when n tends to infinity

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So it will be infinite

slender magnet
exotic sparrow
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I meant option b discarded because iska toh limit infinity ja rha

slender magnet
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kiska.

exotic sparrow
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ncos(1/n) ka

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Option B ka

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And option A bolzano weistras se hat gya na?

slender magnet
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kiska baat kar rahe ho

exotic sparrow
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Vc aa sakte bhai?

exotic sparrow
slender magnet
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but aarha

exotic sparrow
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Type kr lio

slender magnet
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yeah

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okay

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konse wale batana specifically

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theek he

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dikh nahi rha

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quality kharab he

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ok

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nope

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mhm

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okay

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ha

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konse proof specifically batana

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sin(n)/n ?

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are sin(n) ka absolute value 1 se kaam nahi he?

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ha

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hogya

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bounded matlab convergent nahi

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bounded monotone matlab convergent

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ha

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tao se padhe ye?

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tao se dekhlo ekbar

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woh analysis pe sabse aasan book he

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😆 english me padhna practice karo

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abhi english level pe lao

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kiuki sabkuch ke liye woh chaiye

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ok

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unke proof dekhe?

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$a_n \to p,b_n \to q \implies a_nb_n\to pq$

real onyxBOT
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Coffe𝑦

390895986106564609.png
slender magnet
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ye waale

exotic sparrow
slender magnet
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dikhao

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haa ekdam

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definition bohot important he

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ye proof dekha?

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bolzano weirstrass ki proof

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kisika proof samajhna matlab pura samajhna

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me kia batau, tao ke pehle woh sequence khatam karne me 1-1.5 mahine lage

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hojayga tension mat lo

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JAM?

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metric space ha

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metric space real analysis me hi he

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@sullen turret tu bhi real analsysis kar raha he na?

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abhi start kiya?

exotic sparrow
sullen turret
slender magnet
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download krke dekhta

slender magnet
sullen turret
#

kya kaam h jldi bol

slender magnet
#

bata na

sullen turret
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teri mummy

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chup

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mai jaara

slender magnet
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bc bata na

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manhus kisam ke hote ye

slender magnet
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ye sachme 15 saal ka he.

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na

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math pursue karni he toh krlete

slender magnet
slender magnet
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hint deke rakha but yehi he basically proof

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$a_m x_m - aq = a_m(x_m-q) + (a_m-a)q$

real onyxBOT
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Coffe𝑦

390895986106564609.png
slender magnet
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ab a_m aur x_m bounded he na?

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kiuki convergent he

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ha toh aisa ek $M$ he ki $\abs{a_m}, \abs{x_m} <M$

real onyxBOT
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Coffe𝑦

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slender magnet
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(bounded)

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ha dono kuch number se chote he

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unme se bara wala lelo

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-M < a_m < M
-M < x_m < M
ye aaya

slender magnet
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bas a_m ke hi lagalo okay

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haan batao

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konsa

slender magnet
slender magnet
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a_m aur q jo bounded he woh unke saath niche kheecha jayga

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personally ye cheeze tao me utni acchi nahi he aur proofs kuch baare he

slender magnet
real onyxBOT
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Coffe𝑦

390895986106564609.png
slender magnet
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yaha se squeeze theorem lagalo

slender magnet
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bounded to he

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ye zero pe converge bhi karta

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konse

exotic sparrow
slender magnet
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pehle deke rakha ki x_m q pe converge karta aur a_m a pe

slender magnet
#

2 mahine lagbhag lage aadat aane ki

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aur dusre logo ko iske baare me pucha

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but abhi jo bata raha ye dusre book se he