(X,d) be a compact metric space. For every open cover, show there exists ε > 0 such that ∀ ∈ X, B(x,ε) is contained in some member of the cover. 
Let (X,d) be a compact metric space. For every open cover, show there exists ε > 0 such that for every x ∈ X, B(x,ε) is contained in some member of the cover.

My attempt:
(X,d) is compact. Therefore there exists a finite subcover of X.
Any element x in X must lie in some member of the cover, say x ∈ Ui. Otherwise they would not constitute a cover.
Since Ui is open, by definition every point is interior, so there exists ε > 0 such that B(x,ε) is contained in Ui. 
I haven't used the fact the subcover is finite, or the fact X is a metric space rather than just topological space, so I feel my reasoning is flawed. 
Any help is greatly appreciated!

The Wiki proof linked in the comments uses the fact that a continuous function on a compact set reaches its extrema. If you want a proof from scratch and closer to what you are trying to do, here are a few hints:
$1).\ $ Let $\mathcal A$ be an open cover of $X$. For each $x\in X$ there is an open neighborhood $B_{\epsilon_x}(x)$ such that each such $B$ is contained in an element of $\mathcal A$.
$2).\ $ The $B's$ give you $\textit{another}$ open cover of $X$.
$3).\ $ Take a finite subcover of the cover from $2)$ and note that you also get a finite number of $\epsilon_x's$.
$4).\ $ Using the conclusion in $3)$, define $\delta>0$ appropriately to conclude.
