Exercise 9.5 Prove that if F_1,...,F_N⊆R^n is a finite collection of closed sets, then F_1\cup F_2\cup ...\cup F_N Is closed. Hint: First prove the claim for N = 2 using Theorem 9.1. Then use induction. Theorem 9.1. Let F⊆R^n be some set. The following two conditions are equivalent: F is closed. For any convergent sequence 〖(a_k)〗_(k=1)^\infty with a_k in F for all k in N, we must have 〖lim〗_(k->\infty ) a_k in F.