Covering of a compact set

Choose $N>0$ such that $K\subset [-N,N]^n$ and consider collections of disjoint balls with radii $r$ that are centered in $K$. All such balls have the same positive volume and are contained in the set $[-N-r,N+r]^n$ which has finite volume. Hence there is a (finite) upper bound on how many balls such collections can contain. This means that there is a maximal collection, say $\{B(x_1, r), \ldots ,B(x_n, r)\}$, with the above property. If the $B(x_i, 2r)$ did not cover $K$ this would be a direct violation of maximality.


Let $K$ be a compact metric space.

The set $\mathscr X$ of sets $X\subseteq K$ woth the property $$ \forall x,y\in X\colon x\ne y\to d(x,y)\ge 2r$$ is partially ordered by inclusion. The union of a chain in $\mathscr X$ is also $\in\mathscr X$. Hence Zorn's lemma applies and so let $M\in\mathscr X$ be maximal. Suppose there is $a\in K$ with $a\notin\bigcup_{x\in M}B(x,2r) $. Then $M\cup\{a\}\in\mathscr X$, contradicting maximality of $M$.

Finally, by compactness of $K$, there is a finite subset $M_0$ of $M$ such that the $B(x,2r)$ with $x\in M_0$ still cover.

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Real Analysis

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