Real Analysis, Lecture 16: Subsequences, Cauchy Sequences

[@Cemsicles]

it is 2021 now... and this potato cam lecture from ten years ago is still better and more intuitive than the one I am attending in real life.

[@siddharthaganguly9286]

The audience is really something for this series. They ask great questions, and Willie and Jenny's killing it. Savage

[@milksushi6640]

Been watching all these lectures. I believe in every single one so far Willie has managed to get a question in. Well done Willie.

[@siddharthaganguly9286]

Willie's really killing it

[@oddvars]

What was the first excellent question?

[@1468910]

There's a mistake in the proof that lim (s_n)(t_n) = st (where s = lim(s_n) and t = lim(t_n)). Essentially, he's not handling the possibility that one or both of the original limits might be negative. For example, if s_n = t_n = -(10 + 1/n) then s = t = -10. Now given epsilon = 1 (say), we get K = max(-10,-10, 1) = 1, allowing N = 4 (since n >= 4 implies |s_n - s| < 1/3 and |t_n - t| < 1/3). But it's *not* true that |(s_n)(t_n) - st| < epsilon for n >= N since |(s_4)(t_4) - st| = |(41/4)^2 - 100| = 81/16 > 1. The fix is to take K = max(|s|, |t|, 1). (In the example, N would now be 31 and |(s_31)(t_31) - st| = 621/961 < 1.)

[@Fematika]

It actually still works if that is the case, if you think about it.

[@Santino9219]

I just want to thank HMC and professor Su for this lectures, they have been really helpful. However I wonder if for a next time you could use a HD cam, is really hard to read the board with the current definition. Thank you very much.

[@hassanhashemi6478]

subsequences 29:43

[@JaspreetSingh-zp2nm]

Edit: I heard it partly she asked ,is there a limit to how many subsequencial limits you can have, example {2,π,2,π,2,π.....} has two subsequence limit i.e lim{2,2,2...} =2 & lim{π,π,π...}=π, answer to this question I think is uncountable. For example consider set of all rationals put into sequence using arrays you can somehow construct subsequence of sequence of rationals converging to every real number, correct me if I am wrong.

[@SequinBrain]

still figuring this out, but for 2, π, or any other constant, I can't call it "converging" when the limit = 2 & π respectively. So, the limit doesn't converge to 2, it IS 2 for subsequence alpha. To answer the question, assuming you heard correctly, no, there can't be a subsequential limit, since we can assign anything to any sequence. If not, then yes.

[@bentupper4614]

Thanks for posting the question. I agree: I believe a sequence can have arbitrarily - even uncountably - many subsequence with distinct limits. To expand on your example: Let {an} be a sequence of rational numbers. Order them as you would using Cantor's Diagonalization argument. A subsequence with the rational limit p/q can be achieved by choosing from {an} just the terms p/q, 2p/2q, 3p/3q, ... Also, a subsequence with the irrational limit *r* can be constructed by choosing increasing decimal expansions 3, 3.1, 3.14, 3.141, ...

[@frankfurtschoolboy]

I don't think so. Watch the proof again. the point at which he introduces the idea epsilon

[@rootberg]

good lectures, but you should re-upload in better quality. It gets really blocky sometimes making it impossible to read the blackboard for a second now and then.

[@namanrollno5562]

you can refer the notes

[@jaegercuyo115]

great lectures. They are really helping me.

[@deluks917]

What was the question? I cannot here it?

[@焚琴煮鹤-y1t]

I’m always wondering what those phantoms are on the blackboard. Later I found out that it’s the Phantoms of the student

[@heyjack7]

Cauchy Sequence - 56:32

[@0liviashi688]

The lecture is great. It will be a lot better if I can actually see what's in the blackboard

[@lawrenceli9308]

Perfect lecture! Indeed helpful, thank you!

[@TheAbcdane1]

because for each pn, there is a ball around pn which doesn't contain infinitely many points of p? or is this a bad idea and i'm totally off?

[@footstep002]

This maybe very silly question, but at 25:07 he says 'epsilon over k is less than 1 so this becomes less than epsilon over 3', isn't that part have to be s/k not epsilon/k? Because k is max of s and t, so k is greater than or equal to s. In that case it's finally less than epsilon/3. Well less than or equal to epsilon/3 to be more precise...I think. Of course I'm just a viewer who's very poor at math, so my opinion here might be totally wrong. So can someone please verify this?

[@memexd4840]

he probably misspoke, i assume he meant "s over k is less than 1...", not epsilon over k

[@Rambo12zk]

DKM101 What you did is not wrong, but you used the complex identity in real analysis. Although the proof seems somewhat long, it is completely based on real numbers and thus gives more insight.

[@solaris413]

51:00

[@Yeonjun_Choi_]

42:16

[@TheAbcdane1]

does the reverse implication of the compactness and sequentially compactness proof work by contrapositive: (not compact => existence of a sequence that has no convergent subsequences)... if X were not compact, then there would exist an infinite cover of X that would not have a finite cover. Thus, there could exist an infinite sequence in which one element of the sequence were an element in one open set of the cover, such a sequence has no convergent subsequence.

[@2011CatsEyes]

Very helpful, thank you

[@prvnpn216]

cleared few doubts !! thanxx :-)

[@yku01993]

thanks a lot Sir!

[@TheAbcdane1]

sorry, "doesn't contain infinitely many points of the SEQUENCE pn*"

[@zihaojing1120]

HD cam

[@TehFingergunz]

0:00 "NNEEEHHH"

[@Fematika]

Nice.

[@tahseensaad7860]

I really I have got to say an important comment which is from my point of view and I would like to share it with you guys. I realized Proff Su uses words like "something, big thing, one thing, this one, that one, here, that" .A great lecture in this high class, he should avoid using these types of words instead of using the original names indicated to these subjects. It would be better in meaning as well as powerful lecture to fulfill a better understanding and building the "intuition" as he say. I hope he could see my comment and consider that. but no doubt he is great Proff which I admire.

[@krajco252]

lol he is to slow and boring ....