# Tagged Questions

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### Why a sequence $\{x_k\}$ in $S$ such that $||x||>k$ cannot be a convergent subsequence?

This is an excerpt from my text: A set $S \subset \mathbb{R}^n$ is said to be compact if for all sequences of points $\{x_k\}$ such that $x_k\in S$ for each $k$, there exists a subsequence ...
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### Sequence of functions that converges a.e. but not in the $L^1$ norm

How can I construct a sequence of functions that converges a.e. but it does not in the $L^1$ norm?
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### The characteristic function induced by L^1 convergence function

Assume $f_n\in L^1(\Omega)$, $f\in L^1(\Omega)$, and $f_n\to f$ in $L^1(\Omega)$ where $\Omega\subset R^N$ is open. Define $$E_t^n:=\{x\in\Omega, f_n(x)>t\}.$$ Hence we have ...
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### Determining convergence or divergence of series

I am wondering the convergence or divergence following series $$\sum_{n=1}^{\infty} \frac{(-1)^n}{ n+\sin (n)} \\$$ My 1st attempt is 'alternating series test'  But, ...
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### Prove that limit exists and is finite, sum of series, Cauchy.

Prove that $\lim_{n \to \infty} \sum_{k=1}^n \frac{(-1)^k}{k}$ exists and is finite. Attempt: Suppose $\{x_n\}$ is real sequence, and $x_n = \frac{(-1)^k}{k}$. I know if I prove that it is Cauchy, ...
This is a relatively simple problem. I'm just making sure I have the right idea here. I'd like to prove that the sequence $\displaystyle a_n = 1 + \frac{1}{n^{1/3}}$ converges. My proof is: We ...