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Proof of nested interval theorem from bolzano weiertress theorem

score 885 · Answer 1 · 2026-04-11 10:31:58

885

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Proof of nested interval theorem from bolzano weiertress theorem

Published on 11 Apr 2026 - 10:31

score 161 · Answer 2 · 2026-04-14 06:13:44

161

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How to prove that $ \Bigg(\sum^{n}\limits_{k=1}\sqrt{\frac{k-\sqrt{k^{2}-1}}{\sqrt{k(k+1)}}}\Bigg)^{2} \le n\sqrt{\frac{n}{n+1}}$ for $n\ge1$

Published on 14 Apr 2026 - 6:13

score 236 · Answer 3 · 2026-04-14 00:46:23

236

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Given $f$ is strictly increasing, is $f$ bounded?

Published on 14 Apr 2026 - 0:46

score 53 · Answer 4 · 2026-04-12 01:57:12

53

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Find the sup and inf of a set with parameter

Published on 12 Apr 2026 - 1:57

score 34 · Answer 5 · 2026-04-15 08:55:38

34

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Question referring to $\sup(D)$ and $\inf(D)$, for set $D$.

Published on 15 Apr 2026 - 8:55

score 89 · Answer 6 · 2026-04-13 05:31:13

89

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Linear homogenous second order ODE without constant coefficients

Published on 13 Apr 2026 - 5:31

score 155 · Answer 7 · 2026-04-16 12:43:05

155

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How can I prove that $\lim_{n\rightarrow +\infty}\int_0^1(\cos\sqrt x)^n\,dx=0$?

Published on 16 Apr 2026 - 12:43

score 112 · Answer 8 · 2026-04-09 04:27:21

112

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Induction with a recursive sequence

Published on 09 Apr 2026 - 4:27

score 112 · Answer 9 · 2026-04-12 14:56:52

112

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On a new definition of fractional derivative

Published on 12 Apr 2026 - 14:56

score 52 · Answer 10 · 2026-04-13 19:56:11

52

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I have some unclear things in mind related to the mean value theorem for the Riemann integrals.

Published on 13 Apr 2026 - 19:56

score 318 · Answer 11 · 2026-04-17 15:05:57

318

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Proving that $\cos^{-1}\frac{4}{5}+\cos^{-1}\frac{12}{13}=\cos^{-1}\frac{33}{65}$

Published on 17 Apr 2026 - 15:05

score 171 · Answer 12 · 2026-04-11 09:43:40

171

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Linear Mappings and Differentiation

Published on 11 Apr 2026 - 9:43

score 171 · Answer 13 · 2026-04-18 13:48:20

171

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Proving that sequence with given criteria is Cauchy?

Published on 18 Apr 2026 - 13:48

score 114 · Answer 14 · 2026-04-17 20:26:09

114

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Among $2n-1$ irrationals there are $x_1,\dots,x_n$ such that for rationals $a_i\ge0$ with $\sum_{i=1}^{n}a_i>0$, $\sum_{i=1}^{n}a_ix_i$ is irrational

Published on 17 Apr 2026 - 20:26

score 1.4k · Answer 15 · 2026-04-13 21:19:49

1.4k

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Alternative proof to Rudin's "any intersection of closed sets is closed"

Published on 13 Apr 2026 - 21:19

List Question

Proof of nested interval theorem from bolzano weiertress theorem

How to prove that $ \Bigg(\sum^{n}\limits_{k=1}\sqrt{\frac{k-\sqrt{k^{2}-1}}{\sqrt{k(k+1)}}}\Bigg)^{2} \le n\sqrt{\frac{n}{n+1}}$ for $n\ge1$

Given $f$ is strictly increasing, is $f$ bounded?

Find the sup and inf of a set with parameter

Question referring to $\sup(D)$ and $\inf(D)$, for set $D$.

Linear homogenous second order ODE without constant coefficients

How can I prove that $\lim_{n\rightarrow +\infty}\int_0^1(\cos\sqrt x)^n\,dx=0$?

Induction with a recursive sequence

On a new definition of fractional derivative

I have some unclear things in mind related to the mean value theorem for the Riemann integrals.

Proving that $\cos^{-1}\frac{4}{5}+\cos^{-1}\frac{12}{13}=\cos^{-1}\frac{33}{65}$

Linear Mappings and Differentiation

Proving that sequence with given criteria is Cauchy?

Among $2n-1$ irrationals there are $x_1,\dots,x_n$ such that for rationals $a_i\ge0$ with $\sum_{i=1}^{n}a_i>0$, $\sum_{i=1}^{n}a_ix_i$ is irrational

Alternative proof to Rudin's "any intersection of closed sets is closed"

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