Real Analysis I · Topology & Compactness
Compact sets have finite epsilon-nets
Problem
Let $K\subseteq\mathbb R$ be compact and $\varepsilon>0$. Prove there are $x_1,\dots,x_m\in K$ with $K\subseteq\bigcup_{j=1}^mB(x_j,\varepsilon)$.
Hint
The family $\{B(x,\varepsilon):x\in K\}$ is an open cover of $K$.
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