Question

Cauchy criterion involving $\sum_{k=n}^\infty a_n$ for a series

score 48 · Answer 1 · 2022-07-01 09:49:45

48

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Cauchy criterion involving $\sum_{k=n}^\infty a_n$ for a series

Published on 01 Jul 2022 - 9:49

score 53 · Answer 2 · 2022-07-03 17:46:50

53

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What is the error in my proof that every Cauchy sequence is bounded.

Published on 03 Jul 2022 - 17:46

score 49 · Answer 3 · 2022-07-10 19:16:39

49

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$ a_n $ is a Cauchy sequence $ \Rightarrow $ $ a_n^2 $ is a Cauch sequence proof verification

Published on 10 Jul 2022 - 19:16

score 40 · Answer 4 · 2022-07-16 11:39:45

40

Views

Doubt in proving an example of Sequence

Published on 16 Jul 2022 - 11:39

score 39 · Answer 5 · 2022-07-17 21:41:02

39

Views

If $K$ is an unbounded subset of a metric space $(X, d)$, then $K$ contains a non-cauchy sequence

Published on 17 Jul 2022 - 21:41

score 113 · Answer 6 · 2022-07-18 03:08:56

113

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Compact sets are bounded

Published on 18 Jul 2022 - 3:08

score 81 · Answer 7 · 2022-07-18 23:21:04

81

Views

Proof that a compact and unbounded set contains a sequence all subsequences of which are not cauchy

Published on 18 Jul 2022 - 23:21

score 60 · Answer 8 · 2022-07-21 17:49:27

60

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For functions $(f_n)_{n \in \mathbb{N}} \subset L^p, f \in L^p$ prove equivalent condition to convergence in p-norm.

Published on 21 Jul 2022 - 17:49

score 456 · Answer 9 · 2022-07-22 08:59:16

456

Views

Example for sequence which is cauchy but does not converge in L^1 almost everywhere

Published on 22 Jul 2022 - 8:59

score 76 · Answer 10 · 2026-03-28 11:03:39

76

Views

Proving that the map $f:\mathbb R \to \text{Seq}(\mathbb Q)/\sim$ is surjective

Published on 28 Mar 2026 - 11:03

score 132 · Answer 11 · 2022-07-25 16:12:26

132

Views

Proof explanation: If $a_n \leq c \text{ } \forall \text{ } n \in \mathbb{N}$ for $c \in \mathbb{R}$, then $\lim_{n\to\infty} a_n \leq c$

Published on 25 Jul 2022 - 16:12

score 122 · Answer 12 · 2022-07-30 05:19:30

122

Views

A metric space is totally bounded if and only if every sequence has a Cauchy subsequence

Published on 30 Jul 2022 - 5:19

score 65 · Answer 13 · 2026-03-25 22:06:29

65

Views

If the underlying metric is complete, then so is its induced Hausdorff metric

Published on 25 Mar 2026 - 22:06

score 451 · Answer 14 · 2022-08-13 23:17:56

451

Views

Proving that every Cauchy sequence is bounded.

Published on 13 Aug 2022 - 23:17

score 132 · Answer 15 · 2022-08-14 22:30:30

132

Views

Exercise $5.2.1$ - Analysis $1$ by Tao

Published on 14 Aug 2022 - 22:30

List Question

Cauchy criterion involving $\sum_{k=n}^\infty a_n$ for a series

What is the error in my proof that every Cauchy sequence is bounded.

$ a_n $ is a Cauchy sequence $ \Rightarrow $ $ a_n^2 $ is a Cauch sequence proof verification

Doubt in proving an example of Sequence

If $K$ is an unbounded subset of a metric space $(X, d)$, then $K$ contains a non-cauchy sequence

Compact sets are bounded

Proof that a compact and unbounded set contains a sequence all subsequences of which are not cauchy

For functions $(f_n)_{n \in \mathbb{N}} \subset L^p, f \in L^p$ prove equivalent condition to convergence in p-norm.

Example for sequence which is cauchy but does not converge in L^1 almost everywhere

Proving that the map $f:\mathbb R \to \text{Seq}(\mathbb Q)/\sim$ is surjective

Proof explanation: If $a_n \leq c \text{ } \forall \text{ } n \in \mathbb{N}$ for $c \in \mathbb{R}$, then $\lim_{n\to\infty} a_n \leq c$

A metric space is totally bounded if and only if every sequence has a Cauchy subsequence

If the underlying metric is complete, then so is its induced Hausdorff metric

Proving that every Cauchy sequence is bounded.

Exercise $5.2.1$ - Analysis $1$ by Tao

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