Question

If $x$ is a real number, show that there exists a Cauchy sequence of rationals $\{x_n\}$ representing $x$ such that $x_n < x$ for all n.

score 779 · Answer 1 · 2020-01-28 19:48:14

779

Views

If $x$ is a real number, show that there exists a Cauchy sequence of rationals $\{x_n\}$ representing $x$ such that $x_n < x$ for all n.

Published on 28 Jan 2020 - 19:48

score 206 · Answer 2 · 2020-02-02 20:26:51

206

Views

Mixed sequence of convergent sequences converges

Published on 02 Feb 2020 - 20:26

score 82 · Answer 3 · 2020-02-02 21:17:53

82

Views

Limit of sequence in complete space well-defined?

Published on 02 Feb 2020 - 21:17

score 34 · Answer 4 · 2026-03-26 11:01:42

34

Views

Tell if a set $K \subseteq\mathbb R $, closed in $\tau_d$ and bounded by the distance d is necessarily compact in $\tau_d$

Published on 26 Mar 2026 - 11:01

score 60 · Answer 5 · 2026-03-26 12:40:36

60

Views

What is wrong with my solution about the completeness of this metric space?

Published on 26 Mar 2026 - 12:40

score 43 · Answer 6 · 2026-03-26 12:37:23

43

Views

completeness of the metric $[1,2]\times[1,2]\rightarrow \mathbb R:$ $d(x,y)=|x^4-y^4|$

Published on 26 Mar 2026 - 12:37

score 252 · Answer 7 · 2026-03-26 12:38:30

252

Views

Is the metric space of polynomial of degree $\leq1$ complete with the following metric?

Published on 26 Mar 2026 - 12:38

score 364 · Answer 8 · 2020-02-11 02:34:18

364

Views

Proof with Cauchy criterion

Published on 11 Feb 2020 - 2:34

score 41 · Answer 9 · 2026-02-24 07:14:08

41

Views

Converging sequence of functions; real functional analysis

Published on 24 Feb 2026 - 7:14

score 200 · Answer 10 · 2020-02-12 07:17:30

200

Views

Why isn't $B = \{x \in l_{\infty} : \| x \|_{\infty} \leq 1 \}$ a compact metric space?

Published on 12 Feb 2020 - 7:17

score 898 · Answer 11 · 2020-02-14 18:08:23

898

Views

Example of a sequence of $f_n$ converging to $f$ pointwise but not uniformly and its relation with completeness

Published on 14 Feb 2020 - 18:08

score 175 · Answer 12 · 2020-02-16 06:15:27

175

Views

Prove $\{f_{k}\}$ converges uniformly on $[a,b]$.

Published on 16 Feb 2020 - 6:15

score 56 · Answer 13 · 2020-02-17 07:05:22

56

Views

What is the negation of the definition that a metric space is complete?

Published on 17 Feb 2020 - 7:05

score 71 · Answer 14 · 2026-03-26 12:35:43

71

Views

Confusion about the proof of uniform convergence

Published on 26 Mar 2026 - 12:35

score 100 · Answer 15 · 2020-02-17 12:33:22

100

Views

On the classic example of the sequence $f_n=x^n$

Published on 17 Feb 2020 - 12:33

List Question

If $x$ is a real number, show that there exists a Cauchy sequence of rationals $\{x_n\}$ representing $x$ such that $x_n < x$ for all n.

Mixed sequence of convergent sequences converges

Limit of sequence in complete space well-defined?

Tell if a set $K \subseteq\mathbb R $, closed in $\tau_d$ and bounded by the distance d is necessarily compact in $\tau_d$

What is wrong with my solution about the completeness of this metric space?

completeness of the metric $[1,2]\times[1,2]\rightarrow \mathbb R:$ $d(x,y)=|x^4-y^4|$

Is the metric space of polynomial of degree $\leq1$ complete with the following metric?

Proof with Cauchy criterion

Converging sequence of functions; real functional analysis

Why isn't $B = \{x \in l_{\infty} : \| x \|_{\infty} \leq 1 \}$ a compact metric space?

Example of a sequence of $f_n$ converging to $f$ pointwise but not uniformly and its relation with completeness

Prove $\{f_{k}\}$ converges uniformly on $[a,b]$.

What is the negation of the definition that a metric space is complete?

Confusion about the proof of uniform convergence

On the classic example of the sequence $f_n=x^n$

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