 Closed discrete subsets of compact spaces are finite

Let $D$ be a closed discrete subset of a compact space $X$. Closedness makes $D$ compact. Discreteness makes its singletons open in its subspace topology, so the singleton cover has a finite subcover. Thus $D$ is finite. Closedness is essential: the discrete set $\{1/n:n\ge 1\}$ inside the compact interval $[0,1]$ is infinite and has the missing accumulation point zero.