Hi, I think this proof is easy, but would like someone to check my work since sometimes I miss technicalities on these "easy" proofs. Let K1, ..., Kp be compact sets in R^n. Show that union( Kj, j = 1 to p) is a compact set in R^n. Proof. We show that if K1 and K2 are compact then K1 union K2 is compact. Then apply this fact finitely many times to conclude the original statement. I have a theorem: A set E in R^n is compact IFF E is a bounded closed set. Then if K1 and K2 are compact, they are bounded and closed. Then the union is closed. We define a bounded set as a set contained in the open ball B(0, r), where 0 = (0, 0, ..., 0). K1, and K2 bounded implies K1 contained in B(0, r1), and K2 countained in B(0, r2). Then pick r = max(r1, r2). Then K1 union K2 contained in B(0, r). And hence K1 union K2 is bounded. Thus K1 union K2 is a compact set.