Problem to demonstrate that it is compact

Question

Let $X$ be a Normed Vector Space. For any $x\in X$ and $r>0$, let $W:=\{y∈X:∥y−x∥≤r\}$. Prove: $W$ is closed and if $\dim(X)<\infty$ $W$ is compact.

I have no problems show that it is closed, but do not know how to show it is compact. Any suggestions?

Answer 1

Recall that, for finite dimensional spaces, compact sets are the bounded and closed ones. Therefore, after proving that $ W $ is closed, it is enough to check that it is bounded.