**Problem Statement:** Use the open cover definition of compactness to show that any finite union of compact sets is compact.

**Proof:** Let be compact sets in . We wish to show that is a compact set.

Let be an open cover of . Then is an open cover for for every . Since each is compact there is a finite subcover of which covers for each . So for each we may construct , a finite subset of which covers , for each .

I claim that is a finite subcover of .

First note that is a union of finitely many unions of open sets and so it is an open set. Now to verify that covers . Let , then for some . Thus, by construction, since is an open cover of . Therefore, and we may conclude that is a finite subcover of .

Therefore, every open cover of has a finite subcover and so is compact.

**Reflection:** The key in this proof is that we had a finite union of compact sets. From there all we had to do was union up all of their finite subcovers. A finite union of a finite union is simply a finite union of open sets.