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For $n\in \mathbb{N}$ let $U \subseteq \mathbb{R^n} $ be an open set. Show that there exists sequence $K_m$ of compact subsets of $U$ that fulfills two conditions:

  1. $K_m\subseteq K_{m+1} $ for all $m \in \mathbb{N}$
  2. $\bigcup\limits_{m\in\mathbb{N}} {K_m}=U$

I'm really terrible when it comes to proofs related to compactness, I believe I can start by assuming every $K_m$ is bounded and closed because of Heine-Borel theorem but I don't know how to proceed further.

Theo Bendit
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    Hint: I would show this is true first for open balls $U$. Then, I would use the fact that open sets are countable unions of open balls and finite unions of compact sets are compact. – Theo Bendit Jun 28 '21 at 20:47

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