# Tagged Questions

For questions concerning a specific proof, asking for verification, identifying errors, suggestions for improvement, etc.

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### Is this collection of function uniformly equicontinuous? Hints on the proof.

Let $f_n(x)=\frac{1}{n}\cos(e^{nx})$ for $n\in\mathbb{N}$ be a sequence of functions for $x\in[0,1]$. Is it true that $\{f_n\}$ is a uniformly equicontinuous collection of functions? My attempt so ...
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### Using induction on modified inequalities.

Here's the original problem: Prove by induction that $\left(\frac{1}{2}\right) \left(\frac{3}{4}\right) \cdots \left(\frac{2n-1}{2n} \right) \leq \frac{1}{\sqrt{n+1}}$ for all $n \in \mathbb{N}$. ...
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### Proving that the limit of an integral of a series exists

The goal is to show that the following limit exists $$\lim_{T\to\infty} \frac{1}{T}\int_{-T}^T f(x)dx$$ where $$f(x)=\sum_{n=1}^\infty \frac{e^{ia_n x}}{n^2}$$ I already showed that $f$ is bounded ...
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### Closure of Union contains Union of Closures

I'm teaching my self topology using a book I found. This is the second part of a 4 part question. First part is here. I'm trying to prove the following problem from a book I found: Let $X$ be a ...
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### Function with countably many points of discontinuity

Aside from rigor, is this proof correct? Claim. Let $f$ be a function defined on $[0, 1]$ such that $\lim\limits_{y\to a} f(y)$ exists for all $a \in [0, 1]$. Then for any $\epsilon > 0$ there are ...
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### How to show a set is compact using sequential compactness definition?

Let $l^{\infty}$ be the vector space of all bounded sequences $x=(x_n)$ of real numbers with the norm $||x||=\sup_{n\in\mathbb{N}}|x_n|$ and $l^{\infty}$ is complete. I am trying to show that the set ...
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### Looking for feedback on proof (math logic)

$\def\sD{\mathscr{D}} \def\sC{\mathscr{C}} \def\sB{\mathscr{B}} \def\Gm{\Gamma}$ This is a theorem in Mendelson's Intro. to Math. Logic (pg 66, Proposition 2.4). I try to follow his conventions. MP ...
Let $u \in \mathcal{D}'(\Omega)$ and $U \subset \Omega$ open. By definition we say that $u \in \mathcal{E}(U)$ if $\exists u(x) \in \mathcal{E}(U)$ such that \begin{align*} \displaystyle \langle \...