Final answer:
To prove that a closed subset of a compact set is compact, we need to show that every open cover of the closed subset has a finite subcover. This can be done by taking advantage of the properties of closed sets and compact sets.
Step-by-step explanation:
To prove that a closed subset of a compact set is compact, we need to show that every open cover of the closed subset has a finite subcover. Let's assume that we have a closed subset C of a compact set K. Now, let Oα be an open cover of C. Since C is closed, its complement, K - C, is open. And since K is compact, it has a finite subcover, let's say O1, O2, ..., On that covers K. Since O1, O2, ..., On and Oα are open covers of C, their union will be a finite subcover of C. Therefore, a closed subset of a compact set is compact.