Question

Let $(x_n)$ be a bounded sequence and $u=\limsup x_n$. Let E be set of limits of convergent subsequences of $(x_n)$. How do I prove $u \in E$?

score 322 · Answer 1 · 2019-05-24 18:00:10

322

Views

Let $(x_n)$ be a bounded sequence and $u=\limsup x_n$. Let E be set of limits of convergent subsequences of $(x_n)$. How do I prove $u \in E$?

Published on 24 May 2019 - 18:00

score 199 · Answer 2 · 2019-05-29 07:11:58

199

Views

Prove that $\limsup_{n\to\infty}(a_n)\le \sup(a_n)$

Published on 29 May 2019 - 7:11

score 102 · Answer 3 · 2019-05-31 20:12:27

102

Views

If there's only one limit point, what's $\limsup_{n\to \infty} a_n$ and what's $\liminf_{n\to \infty} a_n$?

Published on 31 May 2019 - 20:12

score 278 · Answer 4 · 2019-06-02 11:57:51

278

Views

If $\limsup\limits_{n\rightarrow \infty} a_n=a< \infty$, then $\forall\epsilon>0$ $\exists N\in \mathbb{N}:a_n\leq a+\epsilon$

Published on 02 Jun 2019 - 11:57

score 259 · Answer 5 · 2026-03-25 05:03:02

259

Views

Limit of a power series in $\beta$ multiplied by $(1 - \beta)$

Published on 25 Mar 2026 - 5:03

score 1.4k · Answer 6 · 2019-06-07 12:45:55

1.4k

Views

Radius of convergence of power series $\sum _{n=1} ^{\infty} \frac{n!}{n^n} x^{2n}$

Published on 07 Jun 2019 - 12:45

score 347 · Answer 7 · 2019-06-16 02:55:22

347

Views

Tao Lemma 6.4.13.

Published on 16 Jun 2019 - 2:55

score 194 · Answer 8 · 2019-06-16 21:39:56

194

Views

Proof that $a_n \leq b_n$ implies that $\limsup a_n \leq \limsup b_n$.

Published on 16 Jun 2019 - 21:39

score 54 · Answer 9 · 2019-06-18 13:10:57

54

Views

What is $\lim \sup_{n \to \infty} \sum_{k=1}^{n} (-1)^k \left(\frac{1}{2} -\frac{1}{k} \right)$?

Published on 18 Jun 2019 - 13:10

score 44 · Answer 10 · 2019-06-19 01:13:18

44

Views

ε environment of number 1

Published on 19 Jun 2019 - 1:13

score 127 · Answer 11 · 2019-06-19 23:39:58

127

Views

Given $\limsup \frac{a_n}{b_n} < \infty$. Prove there is a constant M such that $a_n \leq Mb_n$

Published on 19 Jun 2019 - 23:39

score 93 · Answer 12 · 2019-06-20 12:41:27

93

Views

Probability of Brownian Motion

Published on 20 Jun 2019 - 12:41

score 59 · Answer 13 · 2019-06-23 10:16:32

59

Views

Equivalence of a lim sup of functions with a lim inf of sets.

Published on 23 Jun 2019 - 10:16

score 85 · Answer 14 · 2019-06-25 09:17:43

85

Views

A seemingly true claim?

Published on 25 Jun 2019 - 9:17

score 163 · Answer 15 · 2019-07-01 17:01:44

163

Views

$\limsup$ and $\liminf$ of family of sequences with given properties

Published on 01 Jul 2019 - 17:01

List Question

Let $(x_n)$ be a bounded sequence and $u=\limsup x_n$. Let E be set of limits of convergent subsequences of $(x_n)$. How do I prove $u \in E$?

Prove that $\limsup_{n\to\infty}(a_n)\le \sup(a_n)$

If there's only one limit point, what's $\limsup_{n\to \infty} a_n$ and what's $\liminf_{n\to \infty} a_n$?

If $\limsup\limits_{n\rightarrow \infty} a_n=a< \infty$, then $\forall\epsilon>0$ $\exists N\in \mathbb{N}:a_n\leq a+\epsilon$

Limit of a power series in $\beta$ multiplied by $(1 - \beta)$

Radius of convergence of power series $\sum _{n=1} ^{\infty} \frac{n!}{n^n} x^{2n}$

Tao Lemma 6.4.13.

Proof that $a_n \leq b_n$ implies that $\limsup a_n \leq \limsup b_n$.

What is $\lim \sup_{n \to \infty} \sum_{k=1}^{n} (-1)^k \left(\frac{1}{2} -\frac{1}{k} \right)$?

ε environment of number 1

Given $\limsup \frac{a_n}{b_n} < \infty$. Prove there is a constant M such that $a_n \leq Mb_n$

Probability of Brownian Motion

Equivalence of a lim sup of functions with a lim inf of sets.

A seemingly true claim?

$\limsup$ and $\liminf$ of family of sequences with given properties

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