Question

Is the "General" Stolz-Césaro Theorem true for the case $0/0$?

score 26 · Answer 1 · 2022-11-02 12:13:31

26

Views

Is the "General" Stolz-Césaro Theorem true for the case $0/0$?

Published on 02 Nov 2022 - 12:13

score 34 · Answer 2 · 2022-11-02 15:32:16

34

Views

Prove different expression of $\left\{\liminf\limits_{n\to\infty}X_n\geq X\right\}^c$

Published on 02 Nov 2022 - 15:32

score 64 · Answer 3 · 2022-11-02 21:36:38

64

Views

Show that $M\subseteq\bigcup\limits_{m=1}^{\infty}\bigcap\limits_{n=1}^{\infty}\bigcup\limits_{j=n}^{\infty}\left\{ f_j(x)-f(x)>\frac{1}{m}\right\}$

Published on 02 Nov 2022 - 21:36

score 31 · Answer 4 · 2022-11-05 12:17:22

31

Views

Complement of $\left\{x\in\mathbb{R}\mid \liminf\limits_{n\to\infty}f_n(x)\geq f(x)\right\}$

Published on 05 Nov 2022 - 12:17

score 84 · Answer 5 · 2022-11-06 14:41:26

84

Views

Continuous extensions for an inequality

Published on 06 Nov 2022 - 14:41

score 213 · Answer 6 · 2026-03-23 08:31:49

213

Views

Definition of upper semi-continuous functions: limsup or liminf?

Published on 23 Mar 2026 - 8:31

score 118 · Answer 7 · 2022-11-13 08:00:15

Prove that there exists a sequence $x_n\to\infty$ such that $\lim_{n\to\infty} f(x_n) = b$ and $\lim_{n\to\infty} f'(x_n) = 0$

score 134 · Answer 8 · 2022-11-15 18:35:03

134

Views

Is $\liminf\Bbb P(X_nY_n\leq b)\geq\Bbb P(Xy\leq b) $ if $X_n\to X$ in distribution and $Y_n\to y\in\Bbb R$ in probability?

Published on 15 Nov 2022 - 18:35

score 19 · Answer 9 · 2026-03-23 16:50:49

Show $\{x\in M\mid\liminf\limits_{n\to\infty}f_n(x)\geq 1\}\neq\bigcup\limits_{n=1}^{\infty}\bigcap\limits_{k=n}^{\infty}\{x\in M\mid f_n(x)\geq 1\}$

score 93 · Answer 10 · 2022-11-20 19:16:35

93

Views

Proof if $(x_n)$ is bounded sequence, then it converges $\iff$ $\limsup(x_n)=\liminf(x_n)$

Published on 20 Nov 2022 - 19:16

score 100 · Answer 11 · 2022-11-21 13:27:25

100

Views

Definition of $\limsup_{x\to 0^+}f(x)$

Published on 21 Nov 2022 - 13:27

score 56 · Answer 12 · 2022-11-22 09:55:57

56

Views

Determining $\lim\sup_{n→∞} q_n$ and $\lim\inf_{n→∞}q_n$ for a countable set $\mathbb Q ∩ [0, 1]$

Published on 22 Nov 2022 - 9:55

score 146 · Answer 13 · 2022-11-29 16:49:40

146

Views

Trouble on proof step of: $\lim_{n\rightarrow+\infty}a_n=+\infty \Longrightarrow \liminf_{n\rightarrow+\infty}a_n=+\infty$

Published on 29 Nov 2022 - 16:49

score 84 · Answer 14 · 2022-12-01 16:33:03

84

Views

Show that $\binom{2n}{n} = \frac{4^n}{\sqrt{\pi n}} \left(1+\mathcal{O}\bigl(\frac{1}{n}\bigr)\right)$

Published on 01 Dec 2022 - 16:33

score 51 · Answer 15 · 2022-12-06 11:52:47

51

Views

Trouble with proving that if $(a_n)_n,(b_n)_n$ and $a_n \le b_n \forall n \ge \nu$ then $\liminf_n a_n \le \liminf_n b_n$

Published on 06 Dec 2022 - 11:52

List Question

Is the "General" Stolz-Césaro Theorem true for the case $0/0$?

Prove different expression of $\left\{\liminf\limits_{n\to\infty}X_n\geq X\right\}^c$

Show that $M\subseteq\bigcup\limits_{m=1}^{\infty}\bigcap\limits_{n=1}^{\infty}\bigcup\limits_{j=n}^{\infty}\left\{ f_j(x)-f(x)>\frac{1}{m}\right\}$

Complement of $\left\{x\in\mathbb{R}\mid \liminf\limits_{n\to\infty}f_n(x)\geq f(x)\right\}$

Continuous extensions for an inequality

Definition of upper semi-continuous functions: limsup or liminf?

Prove that there exists a sequence $x_n\to\infty$ such that $\lim_{n\to\infty} f(x_n) = b$ and $\lim_{n\to\infty} f'(x_n) = 0$

Is $\liminf\Bbb P(X_nY_n\leq b)\geq\Bbb P(Xy\leq b) $ if $X_n\to X$ in distribution and $Y_n\to y\in\Bbb R$ in probability?

Show $\{x\in M\mid\liminf\limits_{n\to\infty}f_n(x)\geq 1\}\neq\bigcup\limits_{n=1}^{\infty}\bigcap\limits_{k=n}^{\infty}\{x\in M\mid f_n(x)\geq 1\}$

Proof if $(x_n)$ is bounded sequence, then it converges $\iff$ $\limsup(x_n)=\liminf(x_n)$

Definition of $\limsup_{x\to 0^+}f(x)$

Determining $\lim\sup_{n→∞} q_n$ and $\lim\inf_{n→∞}q_n$ for a countable set $\mathbb Q ∩ [0, 1]$

Trouble on proof step of: $\lim_{n\rightarrow+\infty}a_n=+\infty \Longrightarrow \liminf_{n\rightarrow+\infty}a_n=+\infty$

Show that $\binom{2n}{n} = \frac{4^n}{\sqrt{\pi n}} \left(1+\mathcal{O}\bigl(\frac{1}{n}\bigr)\right)$

Trouble with proving that if $(a_n)_n,(b_n)_n$ and $a_n \le b_n \forall n \ge \nu$ then $\liminf_n a_n \le \liminf_n b_n$

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