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 237 · Answer 1 · 2026-04-16 04:47:02

237

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 16 Apr 2026 - 4:47

score 113 · Answer 2 · 2026-04-12 08:08:55

113

Views

Mean of two numbers by infinite sequences

Published on 12 Apr 2026 - 8:08

score 636 · Answer 3 · 2026-04-13 20:10:40

636

Views

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

Published on 13 Apr 2026 - 20:10

score 108 · Answer 4 · 2026-04-09 08:56:22

108

Views

Prove $\liminf$ definition

Published on 09 Apr 2026 - 8:56

score 138 · Answer 5 · 2026-04-15 18:38:17

138

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 15 Apr 2026 - 18:38

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 98 · Answer 7 · 2026-04-13 16:14:58

98

Views

Lim inf of addition of sequences

Published on 13 Apr 2026 - 16:14

score 437 · Answer 8 · 2026-04-17 09:52:30

437

Views

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

Published on 17 Apr 2026 - 9:52

score 469 · Answer 9 · 2026-04-15 12:54:31

469

Views

A question about limsup of the sequence of the average

Published on 15 Apr 2026 - 12:54

score 20 · Answer 10 · 2026-04-15 17:07:01

20

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 15 Apr 2026 - 17:07

score 236 · Answer 11 · 2026-04-16 03:54:53

236

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 16 Apr 2026 - 3:54

score 301 · Answer 12 · 2026-04-16 22:47:36

301

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

Published on 16 Apr 2026 - 22:47

score 896 · Answer 13 · 2026-04-11 21:00:59

896

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 11 Apr 2026 - 21:00

score 108 · Answer 14 · 2026-04-17 05:33:17

108

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 17 Apr 2026 - 5:33

score 56 · Answer 15 · 2026-04-12 12:22:15

56

Views

Divergence of limit of function

Published on 12 Apr 2026 - 12:22

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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