List questions from proof-writing

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Going from local to global definition in proving sets are manifolds

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Finding the supremum of $\left\{n^{\frac{1}{n}}\;\middle\vert\;n\in\mathbb{N}\right\}$

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Internal direct sum of kernel of surjective homomorphism and cyclic subgroup

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Prove that all subsets of countable sets are countable

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Prove that all finite sets are countable

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Example for Hilbert quote

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How to prove that $\left\{\frac{1}{n^{2}}\right\}$ is Cauchy sequence

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Rigorous proof of a linear algebra theorem

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Proving the limit $\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n=e$

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Can "and" be Substituted with "+" in proofs

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Autonomous ODE $\dot{x}=f(x)$: $\lim_{t\rightarrow\infty}x(t)=x^\Rightarrow f(x^)=0$

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ODE $f:\text{<0 if $tx > 0$},\text{>0 if $tx<0$}$; show that $x(t)\equiv0$ is the only solution to $\dot{x}=f(t,x)\hspace{0,3cm}, x(0)=0$

Published on 17 Apr 2026 - 10:09

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Proof that $ \frac{3\pi}{8}< \int_{0}^{\pi/2} \cos{\sin{x}} dx < \frac{49\pi}{128}$

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Showing that the set $\{e^{-x}: x > 0, x\in \mathbb{R}\}$ is neither open or closed rigourously.

Published on 16 Apr 2026 - 8:14

List Question

Proving divisibility of integers

Going from local to global definition in proving sets are manifolds

Finding the supremum of $\left\{n^{\frac{1}{n}}\;\middle\vert\;n\in\mathbb{N}\right\}$

Internal direct sum of kernel of surjective homomorphism and cyclic subgroup

Prove that all subsets of countable sets are countable

Prove that all finite sets are countable

Example for Hilbert quote

How to prove that $\left\{\frac{1}{n^{2}}\right\}$ is Cauchy sequence

Rigorous proof of a linear algebra theorem

Proving the limit $\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n=e$

Can "and" be Substituted with "+" in proofs

Autonomous ODE $\dot{x}=f(x)$: $\lim_{t\rightarrow\infty}x(t)=x^\Rightarrow f(x^)=0$

ODE $f:\text{<0 if $tx > 0$},\text{>0 if $tx<0$}$; show that $x(t)\equiv0$ is the only solution to $\dot{x}=f(t,x)\hspace{0,3cm}, x(0)=0$

Proof that $ \frac{3\pi}{8}< \int_{0}^{\pi/2} \cos{\sin{x}} dx < \frac{49\pi}{128}$

Showing that the set $\{e^{-x}: x > 0, x\in \mathbb{R}\}$ is neither open or closed rigourously.

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