183 reputation
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bio website sprajagopal.github.io
location
age 21
visits member for 1 year, 4 months
seen 5 hours ago

I'm a Junior Research Fellow in IIT Bombay, currently working on the Witsenhausen counterexample.


6h
comment Why is that *any* union of open sets is open but only *finitely many* intersections of open sets is open?
I'm accepting this answer even though the other answers are more mathematically rigorous, because the example really clears up my doubts and helps visualize the concept.
8h
accepted Why is that *any* union of open sets is open but only *finitely many* intersections of open sets is open?
1d
comment Why is that *any* union of open sets is open but only *finitely many* intersections of open sets is open?
Really? Definition is all this matter is about? I will leave the question for some more time. You could delete it if you feel this isn't worth the time.
1d
comment Why is that *any* union of open sets is open but only *finitely many* intersections of open sets is open?
I'm not sure I understand. Do you mean that the definition of intersections is defined only for finitely many sets?
1d
asked Why is that *any* union of open sets is open but only *finitely many* intersections of open sets is open?
2d
answered Prove that subsequence converges to limsup
2d
comment Prove that this sequence diverges to infinity.
@BrunoJoyal, I'm trying to understand that. So, divergent simply means that limit does not extend, but converging to infinity means that it does converge in extended real systems. Am I right?
2d
awarded  Curious
Aug
19
comment Prove that subsequence converges to limsup
Question related to your answer: the definition of lim sup is that it is the limit of the sequence defined as $v_N = \sup{s_i:i>N}$. You could from this that $|v_{n_1} - \alpha| < \epsilon$ and go on to choose some $\epsilon$. But you have rather said that this $v_{n_1}$ necessarily belongs to the sequence $s_n$. That is equivalent to saying that the $\sup$ of a set exists in it, which may or may not be true. Also, isn't your proof essentially selecting a subsequence $s_{n_k}$ from the set $A_N = \{s_i: i > N\}$ so that: $\lim \sup {A_N} \geq s_{n_k} \geq \lim \inf A_N$
Aug
19
comment Why are all convergent sequences necessarily Cauchy?
Understood. Thank you. I made the mistake of assuming that the Cauchy requirement was supposed to hold for every $N$, but it is supposed to hold for every $\epsilon>0$.
Aug
19
comment Why are all convergent sequences necessarily Cauchy?
Understood. Thank you. I thought the Cauchy definition was supposed to hold for every $N$. Rather it is supposed to hold $\forall \epsilon>0$,
Aug
19
accepted Why are all convergent sequences necessarily Cauchy?
Aug
19
comment Why are all convergent sequences necessarily Cauchy?
@Jik, thank you, I have corrected it.
Aug
19
revised Why are all convergent sequences necessarily Cauchy?
mistake corrected
Aug
19
comment Why are all convergent sequences necessarily Cauchy?
Yes, I was wrong about the $\epsilon$ part. Thank you. But the question really is not the proof. I understand the proof. I don't understand the consequence. I have added some comments in the bottom answers. If you could read them and answer that too, it would be great.
Aug
19
comment Why are all convergent sequences necessarily Cauchy?
This is my counterexample that I'm trying to understand. I have a photograph. Here, the y-axis is $s_n$ and the x-axis is the indices. Is this sequence Cauchy? If it is, then the middle part does not conform to the necessity of Cauchy sequences, and yet this sequence does converge in the end.
Aug
19
comment Why are all convergent sequences necessarily Cauchy?
But then, that happens only at the ending. The middle part does not have elements that close up. In fact, they diverge.
Aug
19
comment Why are all convergent sequences necessarily Cauchy?
Alright, I have a photograph. Here, the y-axis is $s_n$ and the x-axis is the indices. Is this sequence Cauchy?
Aug
19
comment Why are all convergent sequences necessarily Cauchy?
It's not entirely clear. I'm trying to draw a picture so that I can specifically ask you my doubt. Lemme try and photograph one from my notebook.
Aug
19
comment Why are all convergent sequences necessarily Cauchy?
The question is not the proof. I wanted to know where my understanding of it is wrong. I had already supplied the same proof in the question.