Let $K$ be a compact metric space. Let $\{U_{i}\}$ be an open covering of $K$. Prove that there exists a number $j>0$ such that any ball of radius $j$ is contained in some $U_{i}$.
Here is my attempt:
Let $\{U_{i}\}$ be a covering and let $\{U_{j}\}$ be a finite subcover. Let $j$ be the minimal radius from the balls of the finite subcover. This is the number we are looking for
Is it ok?