Question

Let $x_1,x_2,\cdots$ be a sequence of real numbers such that $|x_n| \leq 1/2^n$, and set $y_n = x_1 + \cdots + x_n$.

score 470 · Answer 1 · 2022-09-19 15:42:37

470

Views

Let $x_1,x_2,\cdots$ be a sequence of real numbers such that $|x_n| \leq 1/2^n$, and set $y_n = x_1 + \cdots + x_n$.

Published on 19 Sep 2022 - 15:42

score 53 · Answer 2 · 2022-09-20 13:06:29

53

Views

How to prove the space $F = \{g(x) = \int_{-c}^c h(w)e^{(\sigma+i w)x}: h(w)\in L^2[-c,c]\}$ is a Banach space with $l_2$-norm or $l_\infty$-norm.

Published on 20 Sep 2022 - 13:06

score 146 · Answer 3 · 2022-09-22 02:47:57

146

Views

Given $X_n$ is Cauchy in $\mathbb{R}$. Is $\frac{1}{X_n}$ Cauchy?

Published on 22 Sep 2022 - 2:47

score 63 · Answer 4 · 2022-09-23 00:00:14

63

Views

Does my alternative proof of Bolzano-Weierstraß make sense, or am I missing something?

Published on 23 Sep 2022 - 0:00

score 45 · Answer 5 · 2026-03-29 04:33:55

45

Views

Prove a sequence of Weak Solution is Cauchy

Published on 29 Mar 2026 - 4:33

score 124 · Answer 6 · 2022-09-25 19:04:04

124

Views

If $(x_n)$ is a Cauchy sequence that does not tend to zero, prove that $\left(\frac{x_n}{\|x_n\|}\right)$ is Cauchy

Published on 25 Sep 2022 - 19:04

score 43 · Answer 7 · 2026-04-07 17:15:39

43

Views

Help in understanding demonstration: every Cauchy sequence is bounded

Published on 07 Apr 2026 - 17:15

score 65 · Answer 8 · 2022-09-30 16:33:44

65

Views

Is $\sin^2(p\pi+\frac{\pi}{2})$ a Cauchy sequence?

Published on 30 Sep 2022 - 16:33

score 166 · Answer 9 · 2026-03-30 17:05:33

166

Views

How to show that the cartesian product of Hilbert spaces is a Hilbert space?

Published on 30 Mar 2026 - 17:05

score 41 · Answer 10 · 2022-10-03 03:00:58

41

Views

Show that ${b_n}$ is convergent sequence

Published on 03 Oct 2022 - 3:00

score 94 · Answer 11 · 2022-10-04 08:41:24

94

Views

How prove the function could extend continuously to completion of metric spaces is equivalent to the function maps Cauchy sequence to Cauchy sequence?

Published on 04 Oct 2022 - 8:41

score 93 · Answer 12 · 2026-04-06 11:38:44

93

Views

Prove or disprove: If $a_n$ is Cauchy, is $\lfloor a_n \rfloor$ Cauchy?

Published on 06 Apr 2026 - 11:38

score 92 · Answer 13 · 2022-10-09 20:19:15

92

Views

If $ (a_n)$ is a Cauchy sequence, then $ \arctan(a_n)$ is a Cauchy sequence.

Published on 09 Oct 2022 - 20:19

score 51 · Answer 14 · 2026-03-29 07:40:22

51

Views

Let $x_n, a_n$ sequences. Prove $\sum_{n=1} ^{\infty} a_nx_n$ converges when $\sum_{n=1} ^{\infty} ||x_n||$ conv. & $\limsup_{n\to\infty}|a_n|<\infty$

Published on 29 Mar 2026 - 7:40

score 40 · Answer 15 · 2026-03-30 18:43:07

40

Views

Showing that the metric $d(n, m) = 2^{-r}$ if $|n - m| = 2^{r}t$ with $t$ odd if $n \neq m$ and $d_2(n, n) = 0$ on $\mathbb Z$ is not complete

Published on 30 Mar 2026 - 18:43

List Question

Let $x_1,x_2,\cdots$ be a sequence of real numbers such that $|x_n| \leq 1/2^n$, and set $y_n = x_1 + \cdots + x_n$.

How to prove the space $F = \{g(x) = \int_{-c}^c h(w)e^{(\sigma+i w)x}: h(w)\in L^2[-c,c]\}$ is a Banach space with $l_2$-norm or $l_\infty$-norm.

Given $X_n$ is Cauchy in $\mathbb{R}$. Is $\frac{1}{X_n}$ Cauchy?

Does my alternative proof of Bolzano-Weierstraß make sense, or am I missing something?

Prove a sequence of Weak Solution is Cauchy

If $(x_n)$ is a Cauchy sequence that does not tend to zero, prove that $\left(\frac{x_n}{\|x_n\|}\right)$ is Cauchy

Help in understanding demonstration: every Cauchy sequence is bounded

Is $\sin^2(p\pi+\frac{\pi}{2})$ a Cauchy sequence?

How to show that the cartesian product of Hilbert spaces is a Hilbert space?

Show that ${b_n}$ is convergent sequence

How prove the function could extend continuously to completion of metric spaces is equivalent to the function maps Cauchy sequence to Cauchy sequence?

Prove or disprove: If $a_n$ is Cauchy, is $\lfloor a_n \rfloor$ Cauchy?

If $ (a_n)$ is a Cauchy sequence, then $ \arctan(a_n)$ is a Cauchy sequence.

Let $x_n, a_n$ sequences. Prove $\sum_{n=1} ^{\infty} a_nx_n$ converges when $\sum_{n=1} ^{\infty} ||x_n||$ conv. & $\limsup_{n\to\infty}|a_n|<\infty$

Showing that the metric $d(n, m) = 2^{-r}$ if $|n - m| = 2^{r}t$ with $t$ odd if $n \neq m$ and $d_2(n, n) = 0$ on $\mathbb Z$ is not complete

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