TITLE: Subset of totally bounded set is totally bounded.
QUESTION [10 upvotes]: Suppose  $(X,d)$ is a metric space where $A\subseteq X$ is totally covered i.e. $\forall \epsilon >0$ $\exists \{a_1,a_2,\ldots,a_m\}\in A $ s.t $A\subseteq \bigcup_{1,\ldots,m}B(a_i,\epsilon)$. Given that $B\subseteq A$ show that $B$ is totally covered. 
I can't seem to show  - with the given definition - that the centers of the balls would now come from $B.$

REPLY [4 votes]: Suppose $B \subseteq A$ and let $\varepsilon > 0$. Find a finite covering of $A$ by balls of radius $\frac{\varepsilon}{2}$, say
$$B_{\varepsilon/2}(a_1),\ \dots,\ B_{\varepsilon/2}(a_n)$$
where $a_1,\dots,a_n \in A$.
By relabelling the elements $a_1,\dots,a_n$ if necessary, you can assume that the first $k$ of these balls contains an element of $B$, and the remaining $n-k$ do not. For each $1 \le i \le k$, choose some $b_i \in B_{\varepsilon/2}(a_i)$, and note that $B_{\varepsilon}(b_1),\dots,B_{\varepsilon}(b_k)$ is a cover of $B$.