Say that $X$ is a compact subset of a metric space. Is it true that one can find a finite collection of balls of radius $2r$ for a fixed $r> 0$ such that any ball of radius $r$ with center in $X$ is contained in a ball in this collection?
Edit: To clarify, I want to know for any $B_r(x)$ with $x\in X$, is there a finite collection $\{B_{2r}(x_i)\}$ such that $B_r(x)$ is contained in a single $B_{2r}(x_i)$.
The set of all $B(x,r)$, $x \in X$ covers $X$, which is compact, so finitely many, say $x_1,\ldots,x_n$ centres suffice to cover $X$. If now $x \in X$ for some $x_i$, $x \in B(x_i, r)$ so $d(x,x_i) < r$ and so if $d(x, y) < r$ as well, for some $y$, we have $$d(x_i,y) \le d(x_i, x) + d(x,y) < r+r=2r \text{ so } y \in B(x_i, 2r)$$
And it follows that $B(x,r) \subseteq B(x_i, 2r)$ for that $i$. As $x$ was arbitrary, the $x_i, i=1,\ldots,n$ thus are as requested.