The question is like this.
If $(B_j)_{j=1}^\infty$ is a sequence of open balls that covers a compact subset $K \subset \mathbb R^ n$, prove that there is a positive number $\epsilon$ such that each ball with radius $\epsilon$ and center $x\in K$ is contained in one of the balls $(B_j)_{j=1}^\infty$.
I know this is correct but I am not so clear how to construct the $\epsilon$ we want here, so I tried to prove by contradiction. Then we want to find the negation of the statement "$\exists \epsilon\gt0$ such that $\forall B_\epsilon(x)$, $\exists 1\le k\le n$ with $B_\epsilon(x)\subset B_k$". Since $K$ is compact, $(B_j)_{j=1}^n$ can be its open cover.
Suppose it is not true. Then $\forall \epsilon\gt0$, $\exists B_\epsilon(x)$ such that $\forall 1\le k\le n$, $B_\epsilon(x)\cap (B_k)^c\neq\emptyset$.
First of all, is this negation correct? Secondly, how do I reach a contradiction with this negation?
The negation of you want to prove is$$(\forall\varepsilon>0)(\exists x\in K)(\forall n\in\mathbb N):B_\varepsilon(x)\not\subset B_n.$$In particular,$$(\forall m\in\mathbb N)(\exists x_m\in K)(\forall n\in\mathbb N):B_{1/m}(x)\not\subset B_n.$$This is impossible, because, since $K$ is compact, the sequence $(x_m)_{m\in\mathbb N}$ has a convergent subsequence and the limit $x$ of such a subsequence belongs to some $B_n$, which is an open set. Can you take it from here?