Question

Prove that $\varlimsup_{n\to\infty}(\max\{x_n,y_n\})=\max\{\varlimsup_{n\to\infty}x_n,\,\varlimsup_{n\to\infty}y_n\}$

score 234 · Answer 1 · 2018-10-23 18:36:15

234

Views

Prove that $\varlimsup_{n\to\infty}(\max\{x_n,y_n\})=\max\{\varlimsup_{n\to\infty}x_n,\,\varlimsup_{n\to\infty}y_n\}$

Published on 23 Oct 2018 - 18:36

score 110 · Answer 2 · 2018-10-23 18:43:22

110

Views

Mean of two numbers by infinite sequences

Published on 23 Oct 2018 - 18:43

score 633 · Answer 3 · 2018-10-25 04:30:48

633

Views

Is $\liminf \limsup$ always greater than or equal to $\limsup \liminf$?

Published on 25 Oct 2018 - 4:30

score 105 · Answer 4 · 2018-10-27 10:54:25

105

Views

Prove $\liminf$ definition

Published on 27 Oct 2018 - 10:54

score 135 · Answer 5 · 2018-10-27 17:16:44

135

Views

Show $x_n \in \left[\liminf(x_n) - K, \limsup(x_n) + K\right] $ for all $n \geq N$ (some natural number N)

Published on 27 Oct 2018 - 17:16

score 294 · Answer 6 · 2026-03-25 17:36:16

294

Views

Moment generating function is finite given limsup property

Published on 25 Mar 2026 - 17:36

score 95 · Answer 7 · 2018-11-01 03:56:08

95

Views

Lim inf of addition of sequences

Published on 01 Nov 2018 - 3:56

score 434 · Answer 8 · 2018-11-01 14:52:37

434

Views

Proving $\limsup (A_n) \cap \limsup (A^{c}_{n}) = \limsup (A_n \cap A^{c}_{n+1})$

Published on 01 Nov 2018 - 14:52

score 466 · Answer 9 · 2018-11-01 22:29:54

466

Views

A question about limsup of the sequence of the average

Published on 01 Nov 2018 - 22:29

score 18 · Answer 10 · 2026-03-23 05:57:11

18

Views

Implications of $a_l\not\in\{x_k:k\ge l\}$ where $a_n:=\sup\{x_k:k\ge n\},\forall n\in\mathbf{N}$.

Published on 23 Mar 2026 - 5:57

score 233 · Answer 11 · 2018-11-02 14:36:42

233

Views

Proving $x_k-\left(\limsup_{n\to\infty} x_n\right)<\epsilon,\forall k\ge M$ for some $M\in\mathbf{N}$

Published on 02 Nov 2018 - 14:36

score 298 · Answer 12 · 2018-11-03 18:01:17

298

Views

Example of $\lim\inf x_n+\lim \inf y_n<\lim \inf(x_n+y_n)$

Published on 03 Nov 2018 - 18:01

score 893 · Answer 13 · 2018-11-04 06:03:51

893

Views

Suppose $F$ is continuous on $[a,b]$.Show that $D^+F(x)=\limsup_{h\to 0\,h\gt0} \frac{F(x+h)-F(x)}{h} $is measurable.

Published on 04 Nov 2018 - 6:03

score 105 · Answer 14 · 2018-11-08 12:42:36

105

Views

Sequence $(f_n) \to f_n$ on $S \subseteq \mathbb{R}$ converges uniformly iff $\lim_{n \to \infty} \sup \{f(x)-f_n(x)| : x\in S\} = 0$

Published on 08 Nov 2018 - 12:42

score 53 · Answer 15 · 2018-11-10 04:32:58

53

Views

Divergence of limit of function

Published on 10 Nov 2018 - 4:32

List Question

Prove that $\varlimsup_{n\to\infty}(\max\{x_n,y_n\})=\max\{\varlimsup_{n\to\infty}x_n,\,\varlimsup_{n\to\infty}y_n\}$

Mean of two numbers by infinite sequences

Is $\liminf \limsup$ always greater than or equal to $\limsup \liminf$?

Prove $\liminf$ definition

Show $x_n \in \left[\liminf(x_n) - K, \limsup(x_n) + K\right] $ for all $n \geq N$ (some natural number N)

Moment generating function is finite given limsup property

Lim inf of addition of sequences

Proving $\limsup (A_n) \cap \limsup (A^{c}_{n}) = \limsup (A_n \cap A^{c}_{n+1})$

A question about limsup of the sequence of the average

Implications of $a_l\not\in\{x_k:k\ge l\}$ where $a_n:=\sup\{x_k:k\ge n\},\forall n\in\mathbf{N}$.

Proving $x_k-\left(\limsup_{n\to\infty} x_n\right)<\epsilon,\forall k\ge M$ for some $M\in\mathbf{N}$

Example of $\lim\inf x_n+\lim \inf y_n<\lim \inf(x_n+y_n)$

Suppose $F$ is continuous on $[a,b]$.Show that $D^+F(x)=\limsup_{h\to 0\,h\gt0} \frac{F(x+h)-F(x)}{h} $is measurable.

Sequence $(f_n) \to f_n$ on $S \subseteq \mathbb{R}$ converges uniformly iff $\lim_{n \to \infty} \sup \{f(x)-f_n(x)| : x\in S\} = 0$

Divergence of limit of function

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