Final answer:
The union of finitely many compact sets is closed and bounded, therefore, it is also compact.
Step-by-step explanation:
To show that the union of finitely many compact sets is compact, we need to prove that the union is closed and bounded. Let A and B be two compact sets. A set is closed if it contains all its limit points. Since A and B are compact, they are closed and bounded. Let's consider the union of A and B, denoted as A ∪ B.
To prove that A ∪ B is compact, we need to show that it is closed and bounded. To prove that it is bounded, we can use the fact that the union of bounded sets is also bounded. For closure, let's assume a point x belongs to the set of limit points of A ∪ B. Since A and B are closed, x must either belong to A or B. Therefore, x belongs to A ∪ B, making it a closed set. Hence, the union of finitely many compact sets is compact.