Question

Show that $X\subset X\cup\left\{ X\right\}$

score 97 · Answer 1 · 2026-04-16 16:27:09

97

Views

Show that $X\subset X\cup\left\{ X\right\}$

Published on 16 Apr 2026 - 16:27

score 201 · Answer 2 · 2026-04-15 08:33:46

201

Views

$\sum\limits_{m\geq1}\sum\limits_{n\geq1}\frac{(-1)^n}{n^3}\sin\left(\frac{n}{m^2}\right)=\frac{\pi^6}{11340}-\frac{\pi^4}{72}$ Numerical evidence

Published on 15 Apr 2026 - 8:33

score 344 · Answer 3 · 2026-04-11 00:29:45

344

Views

Showing an Integrable function is everywhere discontinuous

Published on 11 Apr 2026 - 0:29

score 179 · Answer 4 · 2026-04-14 18:47:54

179

Views

If domain $A$ of the homomorphism $f$ is a field, and if the range of $f$ has more than one element, then $f$ is injective.

Published on 14 Apr 2026 - 18:47

score 172 · Answer 5 · 2026-04-13 22:30:22

172

Views

Verifying an inner product

Published on 13 Apr 2026 - 22:30

score 109 · Answer 6 · 2026-04-14 07:55:54

109

Views

Is it true that $\gcd(s(p^k), D(p^k)) = 1$?

Published on 14 Apr 2026 - 7:55

score 492 · Answer 7 · 2026-04-16 12:32:23

492

Views

Let $\sum_{k=1}^\infty a_n$ be convergent show that $\sum_{k=1}^\infty n(a_n-a_{n+1})$ converges

Published on 16 Apr 2026 - 12:32

score 125 · Answer 8 · 2026-04-12 20:23:06

125

Views

When a subspace of a space is totally disconnected

Published on 12 Apr 2026 - 20:23

score 3.8k · Answer 9 · 2026-04-17 08:55:24

3.8k

Views

Closed subsets of compact sets are compact (original proof)

Published on 17 Apr 2026 - 8:55

score 471 · Answer 10 · 2026-04-15 05:38:18

471

Views

$U\cap K$ is compact in the subspace topology

Published on 15 Apr 2026 - 5:38

score 103 · Answer 11 · 2026-04-13 04:08:35

103

Views

Proof verification: $d$ and $\tilde{d}$ are topologically equivalent

Published on 13 Apr 2026 - 4:08

score 115 · Answer 12 · 2026-04-13 22:01:53

115

Views

Show that the closure of a connected set is connected.

Published on 13 Apr 2026 - 22:01

score 160 · Answer 13 · 2026-04-14 17:35:37

160

Views

Prove that there exists $\xi \in (-1,1)$ such that $f'''(\xi)=3.$

Published on 14 Apr 2026 - 17:35

score 101 · Answer 14 · 2026-04-17 10:16:18

101

Views

Denote $f(x)=\int_x^{x+1}\cos t^2 {\rm d}t.$Prove $\lim\limits_{x \to +\infty}f(x)=0.$

Published on 17 Apr 2026 - 10:16

score 206 · Answer 15 · 2026-04-16 08:09:45

206

Views

Show that $S_n=1+{x\over1!}+{x^2\over2!}+\cdots+{x^n\over n!}$ converges for $n\in\Bbb N,\ x \in\Bbb R$ without using Taylor series.

Published on 16 Apr 2026 - 8:09

List Question

Show that $X\subset X\cup\left\{ X\right\}$

$\sum\limits_{m\geq1}\sum\limits_{n\geq1}\frac{(-1)^n}{n^3}\sin\left(\frac{n}{m^2}\right)=\frac{\pi^6}{11340}-\frac{\pi^4}{72}$ Numerical evidence

Showing an Integrable function is everywhere discontinuous

If domain $A$ of the homomorphism $f$ is a field, and if the range of $f$ has more than one element, then $f$ is injective.

Verifying an inner product

Is it true that $\gcd(s(p^k), D(p^k)) = 1$?

Let $\sum_{k=1}^\infty a_n$ be convergent show that $\sum_{k=1}^\infty n(a_n-a_{n+1})$ converges

When a subspace of a space is totally disconnected

Closed subsets of compact sets are compact (original proof)

$U\cap K$ is compact in the subspace topology

Proof verification: $d$ and $\tilde{d}$ are topologically equivalent

Show that the closure of a connected set is connected.

Prove that there exists $\xi \in (-1,1)$ such that $f'''(\xi)=3.$

Denote $f(x)=\int_x^{x+1}\cos t^2 {\rm d}t.$Prove $\lim\limits_{x \to +\infty}f(x)=0.$

Show that $S_n=1+{x\over1!}+{x^2\over2!}+\cdots+{x^n\over n!}$ converges for $n\in\Bbb N,\ x \in\Bbb R$ without using Taylor series.

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