Question

How to intuitively understand why for a sequence of a sets $A_n$, $\liminf_{n}A_n \subseteq\limsup_{n}A_n$?

score 274 · Answer 1 · 2016-09-20 22:38:31

274

Views

How to intuitively understand why for a sequence of a sets $A_n$, $\liminf_{n}A_n \subseteq\limsup_{n}A_n$?

Published on 20 Sep 2016 - 22:38

score 87 · Answer 2 · 2016-09-21 04:16:28

87

Views

Convergence theorems for improper integrals

Published on 21 Sep 2016 - 4:16

score 125 · Answer 3 · 2016-09-23 03:01:14

125

Views

Find two bounded sequences, $\{s_n\}_{n \in \mathbb N}$ and $\{t_n\}_{n \in \mathbb N}$, such that $\limsup(s_n+t_n))<\limsup(s_n)+\limsup(t_n)$

Published on 23 Sep 2016 - 3:01

score 29 · Answer 4 · 2016-09-28 20:02:53

29

Views

Simple calcul of a limit superior

Published on 28 Sep 2016 - 20:02

score 655 · Answer 5 · 2016-09-28 20:28:47

655

Views

On upper semi-continuous function and limit superior

Published on 28 Sep 2016 - 20:28

score 62 · Answer 6 · 2016-09-29 13:43:35

62

Views

Prove that $\inf_{t>0}\frac{ω(t)}t=\lim_{t\to\infty}\frac{ω(t)}t$ if $ω:[0,\infty)\toℝ$ be bounded above on every finite interval and subadditive

Published on 29 Sep 2016 - 13:43

score 77 · Answer 7 · 2016-10-01 22:46:37

77

Views

Determine limsup and liminf of the seqeuence $a_n = n\cos \left( \frac{\pi n}{4} \right)$

Published on 01 Oct 2016 - 22:46

score 220 · Answer 8 · 2026-03-26 07:59:09

220

Views

Almost sure convergence of $A_n \to \infty$ implies $P(A_n<N \text{ i.o.})=0$

Published on 26 Mar 2026 - 7:59

score 29 · Answer 9 · 2016-10-09 17:57:40

29

Views

When proving $\liminf(a_n)>L$, proving $\liminf(a_n)>a$ with $a<L$ is enough?

Published on 09 Oct 2016 - 17:57

score 198 · Answer 10 · 2016-10-11 02:22:09

198

Views

$f:\Bbb R \to \Bbb R$ is continuous and bounded then $f$ attains maximum or minimum?

Published on 11 Oct 2016 - 2:22

score 598 · Answer 11 · 2016-10-11 18:03:32

598

Views

Procedure of finding accumulation points of set

Published on 11 Oct 2016 - 18:03

score 753 · Answer 12 · 2016-10-12 12:55:02

753

Views

If $\lim\sup$ of $\{s_n\}$ equals $M$, prove that the $\lim\sup$ of any subsequence $\{s_{n_k}\}$ is $\le M$

Published on 12 Oct 2016 - 12:55

score 59 · Answer 13 · 2016-10-12 22:47:03

59

Views

Compute the limsup and the liminf of the following sequence

Published on 12 Oct 2016 - 22:47

score 360 · Answer 14 · 2016-10-13 03:50:15

360

Views

Prove that $\limsup(a+b) \leq \limsup(a) + \limsup(b)$ and $\liminf(a+b) \geq \liminf(a) + \liminf(b)$

Published on 13 Oct 2016 - 3:50

score 118 · Answer 15 · 2016-10-13 15:32:20

118

Views

Proof that $\forall\epsilon>0 \ ,\ \sum\limits_1^\infty P(|X_n-x|>\epsilon)<\infty\implies X_n\xrightarrow{a.s.} x$

Published on 13 Oct 2016 - 15:32

List Question

How to intuitively understand why for a sequence of a sets $A_n$, $\liminf_{n}A_n \subseteq\limsup_{n}A_n$?

Convergence theorems for improper integrals

Find two bounded sequences, $\{s_n\}_{n \in \mathbb N}$ and $\{t_n\}_{n \in \mathbb N}$, such that $\limsup(s_n+t_n))<\limsup(s_n)+\limsup(t_n)$

Simple calcul of a limit superior

On upper semi-continuous function and limit superior

Prove that $\inf_{t>0}\frac{ω(t)}t=\lim_{t\to\infty}\frac{ω(t)}t$ if $ω:[0,\infty)\toℝ$ be bounded above on every finite interval and subadditive

Determine limsup and liminf of the seqeuence $a_n = n\cos \left( \frac{\pi n}{4} \right)$

Almost sure convergence of $A_n \to \infty$ implies $P(A_n<N \text{ i.o.})=0$

When proving $\liminf(a_n)>L$, proving $\liminf(a_n)>a$ with $a<L$ is enough?

$f:\Bbb R \to \Bbb R$ is continuous and bounded then $f$ attains maximum or minimum?

Procedure of finding accumulation points of set

If $\lim\sup$ of $\{s_n\}$ equals $M$, prove that the $\lim\sup$ of any subsequence $\{s_{n_k}\}$ is $\le M$

Compute the limsup and the liminf of the following sequence

Prove that $\limsup(a+b) \leq \limsup(a) + \limsup(b)$ and $\liminf(a+b) \geq \liminf(a) + \liminf(b)$

Proof that $\forall\epsilon>0 \ ,\ \sum\limits_1^\infty P(|X_n-x|>\epsilon)<\infty\implies X_n\xrightarrow{a.s.} x$

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