Question

Proof verification for $\lim \inf x_n + \lim \sup y_n \le \lim \sup(x_n + y_n)$ for bounded $x_n, y_n$

score 3.4k · Answer 1 · 2026-04-16 11:50:29

3.4k

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Proof verification for $\lim \inf x_n + \lim \sup y_n \le \lim \sup(x_n + y_n)$ for bounded $x_n, y_n$

Published on 16 Apr 2026 - 11:50

score 2.1k · Answer 2 · 2026-04-14 07:43:29

2.1k

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Existence of a subsequence converging to limsup

Published on 14 Apr 2026 - 7:43

score 184 · Answer 3 · 2026-04-16 23:16:41

184

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Prove any number $c \in [a, b]$ is a subsequential limit if $\lim\inf x_n = a$, $\lim \sup x_n = b$, $a\ne b$, $\lim(x_n -x_{n+1})=0$

Published on 16 Apr 2026 - 23:16

score 112 · Answer 4 · 2026-04-11 01:41:38

112

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Clarification on step in proof for Ratio test

Published on 11 Apr 2026 - 1:41

score 202 · Answer 5 · 2026-04-17 10:59:37

202

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Is $\limsup_{n\rightarrow\infty}\sqrt[n]{|a_n|}=\frac{1}{R}=\limsup_{n\rightarrow\infty}|\frac{a_{n+1}}{a_n}|$?

Published on 17 Apr 2026 - 10:59

score 120 · Answer 6 · 2026-04-11 18:08:51

120

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Finding $\limsup_{n\to\infty}\sum_{k=0}^n(-1)^k|\sin k|$

Published on 11 Apr 2026 - 18:08

score 57 · Answer 7 · 2026-04-16 02:47:47

57

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For a convergent sequence $\limsup b_n = \lim b_n$

Published on 16 Apr 2026 - 2:47

score 171 · Answer 8 · 2026-04-14 01:42:16

171

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$\liminf E_{n_0+n} = \liminf\limits E_{n}$ and $\limsup E_{n_0+n} = \limsup E_{n}$ where $E_n$ is a decreasing sequence?

Published on 14 Apr 2026 - 1:42

score 182 · Answer 9 · 2026-04-12 05:28:05

182

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Intuition behind inequality for measure of $\liminf$ and $\limsup$

Published on 12 Apr 2026 - 5:28

score 135 · Answer 10 · 2026-04-17 10:52:27

135

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If $\{a_n\}$ is bounded and non-decreasing, prove that $\liminf b_n = 0$, $b_n = n(a_{n+1} - a_{n})$

Published on 17 Apr 2026 - 10:52

score 61 · Answer 11 · 2026-04-09 06:08:38

61

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Let $f$ be Lebesgueable integrable, then $\lim_{h\to 0}\sup \int_{|x -y| < h}|f(x) - f(y)| = 0$

Published on 09 Apr 2026 - 6:08

score 206 · Answer 12 · 2026-04-16 14:01:15

206

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Stolz-Cesàro $0/0$ case: is $\limsup \frac{a_n}{b_n}\le \limsup\frac{a_{n+1}-a_n}{b_{n+1}-b_n}$?

Published on 16 Apr 2026 - 14:01

score 516 · Answer 13 · 2026-04-15 11:57:30

516

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Walter Rudin "Principles of Mathematical Analysis" Definition 3.16, Theorem 3.17. I cannot understand.

Published on 15 Apr 2026 - 11:57

score 142 · Answer 14 · 2026-04-10 09:11:25

142

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$\limsup$ and integral inequality

Published on 10 Apr 2026 - 9:11

score 237 · Answer 15 · 2026-04-14 12:57:56

237

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About "Principles of Mathematical Analysis" by Walter Rudin Theorem 3.17(a).

Published on 14 Apr 2026 - 12:57

List Question

Proof verification for $\lim \inf x_n + \lim \sup y_n \le \lim \sup(x_n + y_n)$ for bounded $x_n, y_n$

Existence of a subsequence converging to limsup

Prove any number $c \in [a, b]$ is a subsequential limit if $\lim\inf x_n = a$, $\lim \sup x_n = b$, $a\ne b$, $\lim(x_n -x_{n+1})=0$

Clarification on step in proof for Ratio test

Is $\limsup_{n\rightarrow\infty}\sqrt[n]{|a_n|}=\frac{1}{R}=\limsup_{n\rightarrow\infty}|\frac{a_{n+1}}{a_n}|$?

Finding $\limsup_{n\to\infty}\sum_{k=0}^n(-1)^k|\sin k|$

For a convergent sequence $\limsup b_n = \lim b_n$

$\liminf E_{n_0+n} = \liminf\limits E_{n}$ and $\limsup E_{n_0+n} = \limsup E_{n}$ where $E_n$ is a decreasing sequence?

Intuition behind inequality for measure of $\liminf$ and $\limsup$

If $\{a_n\}$ is bounded and non-decreasing, prove that $\liminf b_n = 0$, $b_n = n(a_{n+1} - a_{n})$

Let $f$ be Lebesgueable integrable, then $\lim_{h\to 0}\sup \int_{|x -y| < h}|f(x) - f(y)| = 0$

Stolz-Cesàro $0/0$ case: is $\limsup \frac{a_n}{b_n}\le \limsup\frac{a_{n+1}-a_n}{b_{n+1}-b_n}$?

Walter Rudin "Principles of Mathematical Analysis" Definition 3.16, Theorem 3.17. I cannot understand.

$\limsup$ and integral inequality

About "Principles of Mathematical Analysis" by Walter Rudin Theorem 3.17(a).

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