Question

Define $g(a) := \liminf_{x \to a} f(x)$ for all $a \in X$. Then $g$ is lower semi-continuous

score 71 · Answer 1 · 2026-03-27 03:44:20

71

Views

Define $g(a) := \liminf_{x \to a} f(x)$ for all $a \in X$. Then $g$ is lower semi-continuous

Published on 27 Mar 2026 - 3:44

score 384 · Answer 2 · 2022-07-22 09:56:06

384

Views

Prove $\liminf_{n\to\infty} \left(n^2(4a_n(1-a_{n-1})-1)\right)\leq\frac14$ for any non-negative seqence $\{a_n\}$.

Published on 22 Jul 2022 - 9:56

score 59 · Answer 3 · 2022-07-29 10:28:09

59

Views

limsup proof for finite sequences

Published on 29 Jul 2022 - 10:28

score 58 · Answer 4 · 2022-07-31 11:24:09

58

Views

A query concerning $\limsup$ and $\liminf$ of the sequence $\langle 1-\cos(1/n)\rangle$.

Published on 31 Jul 2022 - 11:24

score 120 · Answer 5 · 2022-08-03 19:37:50

120

Views

limit of limit of sets is a limit of sets

Published on 03 Aug 2022 - 19:37

score 143 · Answer 6 · 2022-08-07 02:23:43

143

Views

limsup of events, and events of limsup of a sequence of random variables

Published on 07 Aug 2022 - 2:23

score 53 · Answer 7 · 2022-08-07 05:11:09

53

Views

proving limsup of example with Definition of Limit Superior from Royden

Published on 07 Aug 2022 - 5:11

score 272 · Answer 8 · 2022-08-14 04:28:45

272

Views

If $(a_n)$ is convergent, then $\liminf (a_n+b_n) = \lim a_n + \liminf b_n$

Published on 14 Aug 2022 - 4:28

score 68 · Answer 9 · 2022-08-22 20:00:52

68

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Consider $a_n = \ln(1 + a_{n-1})$ with $a_0 = 1/2$. Show the limit is $0$ and find the radius of convergence (proof verification)

Published on 22 Aug 2022 - 20:00

score 80 · Answer 10 · 2022-09-03 00:29:11

80

Views

Radius of convergence of $\sum_{n=0}^{\infty} \sin(in) \left( \frac{z}{i+1}\right)^{2n}$

Published on 03 Sep 2022 - 0:29

score 69 · Answer 11 · 2022-09-03 09:59:42

69

Views

Another stuck in proof of Theorem 8.1 of Baby Rudin's PMA

Published on 03 Sep 2022 - 9:59

score 35 · Answer 12 · 2022-09-06 14:43:39

35

Views

Can we choose such subsequences?

Published on 06 Sep 2022 - 14:43

score 47 · Answer 13 · 2022-09-08 04:29:19

47

Views

All limit points of $\{(-1)^k+(-1/k)^k\}$

Published on 08 Sep 2022 - 4:29

score 187 · Answer 14 · 2022-09-08 14:14:35

187

Views

If $t:=\limsup a_n$, then for all $\epsilon>0$, exists $N\in\mathbb{N}$ such that $a_n<t+\epsilon$ for all $n\geq N$

Published on 08 Sep 2022 - 14:14

score 37 · Answer 15 · 2022-09-11 17:31:09

37

Views

Limsup of $A_n = \{ \text{"heads" not seen in first $n$ flips}\}$

Published on 11 Sep 2022 - 17:31

List Question

Define $g(a) := \liminf_{x \to a} f(x)$ for all $a \in X$. Then $g$ is lower semi-continuous

Prove $\liminf_{n\to\infty} \left(n^2(4a_n(1-a_{n-1})-1)\right)\leq\frac14$ for any non-negative seqence $\{a_n\}$.

limsup proof for finite sequences

A query concerning $\limsup$ and $\liminf$ of the sequence $\langle 1-\cos(1/n)\rangle$.

limit of limit of sets is a limit of sets

limsup of events, and events of limsup of a sequence of random variables

proving limsup of example with Definition of Limit Superior from Royden

If $(a_n)$ is convergent, then $\liminf (a_n+b_n) = \lim a_n + \liminf b_n$

Consider $a_n = \ln(1 + a_{n-1})$ with $a_0 = 1/2$. Show the limit is $0$ and find the radius of convergence (proof verification)

Radius of convergence of $\sum_{n=0}^{\infty} \sin(in) \left( \frac{z}{i+1}\right)^{2n}$

Another stuck in proof of Theorem 8.1 of Baby Rudin's PMA

Can we choose such subsequences?

All limit points of $\{(-1)^k+(-1/k)^k\}$

If $t:=\limsup a_n$, then for all $\epsilon>0$, exists $N\in\mathbb{N}$ such that $a_n<t+\epsilon$ for all $n\geq N$

Limsup of $A_n = \{ \text{"heads" not seen in first $n$ flips}\}$

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