Why is $P^n(K)$ compact, where $K$ is a local field?

Question

In Silverman's book AEC, question 7.6 asks to prove $E_0(K)$ has finite index in $E(K)$ for $K$ a local field. For part (a), I know the topology on $P^n(K)$ is the quotient topology on $K^{n+1}$, and the topology on $K^{n+1}$ is induced by the abolute value. However, I do not know how to prove the compactness of $P^n(K)$ in this topology.