Question

If for every sequence $(x_n)$, $x_n \to x_0 \implies (f(x_n))$ is a Cauchy sequence, then is $f$ continuous?

score 167 · Answer 1 · 2021-01-20 10:39:15

167

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If for every sequence $(x_n)$, $x_n \to x_0 \implies (f(x_n))$ is a Cauchy sequence, then is $f$ continuous?

Published on 20 Jan 2021 - 10:39

score 28 · Answer 2 · 2021-01-20 18:24:44

28

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cauchy-preserving metric spaces

Published on 20 Jan 2021 - 18:24

score 46 · Answer 3 · 2021-01-24 05:53:23

46

Views

If sum of LI vectors is Cauchy, do the vectors themselves have to be Cauchy?

Published on 24 Jan 2021 - 5:53

score 362 · Answer 4 · 2021-01-26 10:04:14

362

Views

Is a Cauchy sequence bounded in a metric space if the space is not complete?

Published on 26 Jan 2021 - 10:04

score 26 · Answer 5 · 2021-01-26 14:01:14

26

Views

$d$-Cauchy $\Leftrightarrow \forall \epsilon>0 \exists N\in\mathbb N (\text{diam}(\{x_n:n\ge N\}) \le \epsilon)$

Published on 26 Jan 2021 - 14:01

score 612 · Answer 6 · 2021-01-27 16:57:05

612

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If sequences $x_n$ and $y_n$ in a metric space converge to the same limit, does $d(x_n, y_n)$ converge?

Published on 27 Jan 2021 - 16:57

score 245 · Answer 7 · 2021-02-01 01:36:21

245

Views

Proving every Cauchy sequence in $\mathbb{R}$ is convergent

Published on 01 Feb 2021 - 1:36

score 46 · Answer 8 · 2021-02-01 03:52:17

46

Views

Cauchy product rule to replace "pi" with a different infinite series in a particular ellipse

Published on 01 Feb 2021 - 3:52

score 33 · Answer 9 · 2021-02-01 08:57:00

33

Views

Cauchy sequence: distance from point of convergence

Published on 01 Feb 2021 - 8:57

score 81 · Answer 10 · 2021-02-01 14:26:14

81

Views

Why does the condition $|a_{n+1}-a_n|\le1/n$ not imply the sequence is Cauchy?

Published on 01 Feb 2021 - 14:26

score 122 · Answer 11 · 2026-03-31 10:14:48

122

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Show that the Cauchy distribution, whose density is $f\left(x\right)=\frac{1}{\left[\pi\left(1+x^{2}\right)\right]}$ does not possess finite moments.

Published on 31 Mar 2026 - 10:14

score 201 · Answer 12 · 2026-03-27 06:56:57

201

Views

Do Cauchy Sequences and Dedekind cuts have any real-world applications?

Published on 27 Mar 2026 - 6:56

score 134 · Answer 13 · 2021-02-06 10:04:07

134

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Convergence of Cauchy sequences in $(\ell^1, \|\cdot\|_\infty)$

Published on 06 Feb 2021 - 10:04

score 106 · Answer 14 · 2021-02-08 11:48:25

106

Views

Problem in proving Cauchy sequence - how to tackle inequalities and changing indices [EDITED]

Published on 08 Feb 2021 - 11:48

score 72 · Answer 15 · 2021-02-08 18:06:03

72

Views

Sequences and Cauchy proofs

Published on 08 Feb 2021 - 18:06

List Question

If for every sequence $(x_n)$, $x_n \to x_0 \implies (f(x_n))$ is a Cauchy sequence, then is $f$ continuous?

cauchy-preserving metric spaces

If sum of LI vectors is Cauchy, do the vectors themselves have to be Cauchy?

Is a Cauchy sequence bounded in a metric space if the space is not complete?

$d$-Cauchy $\Leftrightarrow \forall \epsilon>0 \exists N\in\mathbb N (\text{diam}(\{x_n:n\ge N\}) \le \epsilon)$

If sequences $x_n$ and $y_n$ in a metric space converge to the same limit, does $d(x_n, y_n)$ converge?

Proving every Cauchy sequence in $\mathbb{R}$ is convergent

Cauchy product rule to replace "pi" with a different infinite series in a particular ellipse

Cauchy sequence: distance from point of convergence

Why does the condition $|a_{n+1}-a_n|\le1/n$ not imply the sequence is Cauchy?

Show that the Cauchy distribution, whose density is $f\left(x\right)=\frac{1}{\left[\pi\left(1+x^{2}\right)\right]}$ does not possess finite moments.

Do Cauchy Sequences and Dedekind cuts have any real-world applications?

Convergence of Cauchy sequences in $(\ell^1, \|\cdot\|_\infty)$

Problem in proving Cauchy sequence - how to tackle inequalities and changing indices [EDITED]

Sequences and Cauchy proofs

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