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 · 2018-12-14 15:50:52

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 14 Dec 2018 - 15:50

score 2.1k · Answer 2 · 2018-12-16 21:42:34

2.1k

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Existence of a subsequence converging to limsup

Published on 16 Dec 2018 - 21:42

score 181 · Answer 3 · 2018-12-17 13:50:49

181

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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 17 Dec 2018 - 13:50

score 109 · Answer 4 · 2018-12-20 11:06:32

109

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Clarification on step in proof for Ratio test

Published on 20 Dec 2018 - 11:06

score 199 · Answer 5 · 2018-12-29 17:17:20

199

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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 29 Dec 2018 - 17:17

score 117 · Answer 6 · 2019-01-01 16:37:41

117

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

Published on 01 Jan 2019 - 16:37

score 54 · Answer 7 · 2019-01-02 15:33:47

54

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

Published on 02 Jan 2019 - 15:33

score 168 · Answer 8 · 2019-01-04 00:18:20

168

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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 04 Jan 2019 - 0:18

score 179 · Answer 9 · 2019-01-05 15:13:50

179

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

Published on 05 Jan 2019 - 15:13

score 133 · Answer 10 · 2026-03-26 04:36:34

133

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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 26 Mar 2026 - 4:36

score 60 · Answer 11 · 2026-02-23 01:08:25

60

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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 23 Feb 2026 - 1:08

score 203 · Answer 12 · 2019-01-16 14:00:48

203

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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 Jan 2019 - 14:00

score 513 · Answer 13 · 2019-01-19 10:13:44

513

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

Published on 19 Jan 2019 - 10:13

score 139 · Answer 14 · 2019-01-19 18:22:55

139

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$\limsup$ and integral inequality

Published on 19 Jan 2019 - 18:22

score 234 · Answer 15 · 2019-01-20 04:33:11

234

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

Published on 20 Jan 2019 - 4:33

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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