Question

Prove that $A_n \cap B_n \rightarrow A \cap B$

score 423 · Answer 1 · 2026-03-12 08:16:40

423

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Prove that $A_n \cap B_n \rightarrow A \cap B$

Published on 12 Mar 2026 - 8:16

score 153 · Answer 2 · 2026-03-12 12:36:40

153

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$X_n \xrightarrow{n \to \infty} X$ almost surely implies $\mathbb{E}[|X|] \le \liminf_{n \to \infty} \mathbb{E}[| X_n |]$

Published on 12 Mar 2026 - 12:36

score 226 · Answer 3 · 2026-03-12 14:10:57

226

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Is there a sequence $x_n\to+\infty$ such that $\liminf x_{2n}/x_n = 0$?

Published on 12 Mar 2026 - 14:10

score 66 · Answer 4 · 2026-03-11 16:25:04

66

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What is the definition of a limit superior for a set valued mapping?

Published on 11 Mar 2026 - 16:25

score 54 · Answer 5 · 2026-03-11 18:24:38

54

Views

Convergence of a sequence of sets $A_n:=\{1+ \frac{m^2}{n^2}: m \in \mathbb{N} \}$

Published on 11 Mar 2026 - 18:24

score 135 · Answer 6 · 2026-03-12 13:14:29

135

Views

Asymptotic equivalence and lim inf, lim sup

Published on 12 Mar 2026 - 13:14

score 71 · Answer 7 · 2026-03-12 02:13:22

71

Views

Any bounded sequence in $\mathbb R$ has a smallest cluster point $\underline x$ and a greatest cluster point $\overline x$

Published on 12 Mar 2026 - 2:13

score 190 · Answer 8 · 2026-03-12 04:57:25

190

Views

Prove that $\underline{\lim} (x_n y_n) \le x_* y^*$

Published on 12 Mar 2026 - 4:57

score 156 · Answer 9 · 2026-03-11 18:39:07

156

Views

How to prove that $\limsup_{n\to\infty}n M_n\geq 1/\ln 2$ for the following sequence?

Published on 11 Mar 2026 - 18:39

score 288 · Answer 10 · 2026-03-11 21:22:31

288

Views

Proving Properties limsup and liminf

Published on 11 Mar 2026 - 21:22

score 53 · Answer 11 · 2026-03-12 13:42:54

53

Views

Limit of a sequence including an exponential function

Published on 12 Mar 2026 - 13:42

score 234 · Answer 12 · 2026-03-11 22:02:06

234

Views

How to prove $\limsup_{k \to \infty} \sqrt[k]{|a_k|} = \limsup_{k \to \infty} \sqrt[k]{ \left |(k+1)a_{k+1} \right|}$

Published on 11 Mar 2026 - 22:02

score 193 · Answer 13 · 2026-03-11 17:27:07

193

Views

convergence of a sequence of a non-empty subset of $\mathbb R$ (existence)

Published on 11 Mar 2026 - 17:27

score 411 · Answer 14 · 2026-03-11 17:58:20

411

Views

$\lim\sup x_n = \infty$ and $\lim\inf x_n = - \infty$

Published on 11 Mar 2026 - 17:58

score 2.6k · Answer 15 · 2026-03-12 00:06:44

2.6k

Views

If $\lim A_n$ exists, then $P(\lim A_n) = \lim P(A_n)$

Published on 12 Mar 2026 - 0:06

List Question

Prove that $A_n \cap B_n \rightarrow A \cap B$

$X_n \xrightarrow{n \to \infty} X$ almost surely implies $\mathbb{E}[|X|] \le \liminf_{n \to \infty} \mathbb{E}[| X_n |]$

Is there a sequence $x_n\to+\infty$ such that $\liminf x_{2n}/x_n = 0$?

What is the definition of a limit superior for a set valued mapping?

Convergence of a sequence of sets $A_n:=\{1+ \frac{m^2}{n^2}: m \in \mathbb{N} \}$

Asymptotic equivalence and lim inf, lim sup

Any bounded sequence in $\mathbb R$ has a smallest cluster point $\underline x$ and a greatest cluster point $\overline x$

Prove that $\underline{\lim} (x_n y_n) \le x_* y^*$

How to prove that $\limsup_{n\to\infty}n M_n\geq 1/\ln 2$ for the following sequence?

Proving Properties limsup and liminf

Limit of a sequence including an exponential function

How to prove $\limsup_{k \to \infty} \sqrt[k]{|a_k|} = \limsup_{k \to \infty} \sqrt[k]{ \left |(k+1)a_{k+1} \right|}$

convergence of a sequence of a non-empty subset of $\mathbb R$ (existence)

$\lim\sup x_n = \infty$ and $\lim\inf x_n = - \infty$

If $\lim A_n$ exists, then $P(\lim A_n) = \lim P(A_n)$

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