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.