Question

Show that $(a_n)$ with $|a_n - a_m|\leq C |\frac{1}{m}-\frac{1}{n}|$ is convergent

score 176 · Answer 1 · 2021-08-24 20:34:36

176

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Show that $(a_n)$ with $|a_n - a_m|\leq C |\frac{1}{m}-\frac{1}{n}|$ is convergent

Published on 24 Aug 2021 - 20:34

score 67 · Answer 2 · 2021-09-03 00:57:06

67

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A explicit rational sequence that converges to $x\in\mathbb{R}$

Published on 03 Sep 2021 - 0:57

score 291 · Answer 3 · 2026-03-30 05:13:15

291

Views

Completeness for a given metric on the sequences of $0s$ and $1s$

Published on 30 Mar 2026 - 5:13

score 166 · Answer 4 · 2021-09-05 08:10:38

166

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$z_n \rightarrow z$ iff $x_n \rightarrow x$ and $y_n \rightarrow y$

Published on 05 Sep 2021 - 8:10

score 183 · Answer 5 · 2026-03-30 10:56:40

183

Views

How do we define the order relation in $\mathbb{R}$ in the construction via Cauchy sequences?

Published on 30 Mar 2026 - 10:56

score 530 · Answer 6 · 2021-09-11 10:43:24

530

Views

Proving every Cauchy sequence in $\mathbb{R}$ is convergent. Does the following proof work?

Published on 11 Sep 2021 - 10:43

score 183 · Answer 7 · 2021-09-13 09:47:01

183

Views

Uniformly convergence of sequence of functions

Published on 13 Sep 2021 - 9:47

score 1.3k · Answer 8 · 2026-04-05 18:47:20

1.3k

Views

Does a continuous function map Cauchy sequences to Cauchy sequences?

Published on 05 Apr 2026 - 18:47

score 55 · Answer 9 · 2021-09-25 04:00:07

55

Views

When does a sequence $x(n)$ actually achieve the value at the limit?

Published on 25 Sep 2021 - 4:00

score 171 · Answer 10 · 2021-09-25 19:26:33

171

Views

Ex 3.1 of Baby Rudin

Published on 25 Sep 2021 - 19:26

score 643 · Answer 11 · 2026-04-05 18:35:10

643

Views

When does Cauchy continuity imply uniform continuity?

Published on 05 Apr 2026 - 18:35

score 32 · Answer 12 · 2021-09-29 11:55:38

32

Views

If $\{x_n\}$ is a Cauchy sequence, then for all $j \,\, \exists N_j$ such that for $n,m > N_j$ we have $|x_n -x_m| < 2^{-j}$

Published on 29 Sep 2021 - 11:55

score 45 · Answer 13 · 2026-02-24 05:12:30

45

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How to prove in metric space that classical definition of the Cauchy sequence coincides with the entourage definition of the Cauchy sequence?

Published on 24 Feb 2026 - 5:12

score 52 · Answer 14 · 2021-10-03 17:10:59

52

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Existence of Cauchy and convergent sequences

Published on 03 Oct 2021 - 17:10

score 20 · Answer 15 · 2026-04-03 12:58:12

20

Views

Prove limit of recursive sequence using epsilon-N

Published on 03 Apr 2026 - 12:58

List Question

Show that $(a_n)$ with $|a_n - a_m|\leq C |\frac{1}{m}-\frac{1}{n}|$ is convergent

A explicit rational sequence that converges to $x\in\mathbb{R}$

Completeness for a given metric on the sequences of $0s$ and $1s$

$z_n \rightarrow z$ iff $x_n \rightarrow x$ and $y_n \rightarrow y$

How do we define the order relation in $\mathbb{R}$ in the construction via Cauchy sequences?

Proving every Cauchy sequence in $\mathbb{R}$ is convergent. Does the following proof work?

Uniformly convergence of sequence of functions

Does a continuous function map Cauchy sequences to Cauchy sequences?

When does a sequence $x(n)$ actually achieve the value at the limit?

Ex 3.1 of Baby Rudin

When does Cauchy continuity imply uniform continuity?

If $\{x_n\}$ is a Cauchy sequence, then for all $j \,\, \exists N_j$ such that for $n,m > N_j$ we have $|x_n -x_m| < 2^{-j}$

How to prove in metric space that classical definition of the Cauchy sequence coincides with the entourage definition of the Cauchy sequence?

Existence of Cauchy and convergent sequences

Prove limit of recursive sequence using epsilon-N

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