Could someone please check my work

1. If S1 , and S2 , are compact subsets of R , prove that S1 U S2 is compact. Suppose S1 and $2 are compact subsets of R . By Theorem 3.5.5 (Heine Borel), since S1 and S2 are compact, they are closed and bounded. By Corollary 3.4.11(b), the union of the closed sets, S1 and S2 is closed. Since S1 and $2 are bounded, they each have a lower bound and an upper bound, namely an infimum and a supremum by the definitions of infimum, supremum, and boundedness. Let x = inf( S1) and y = inf( $2) . x E S1 by the definition of infimum of S1 and y E S2 by the definition of infimum of $2 # x, y E S1 U S2 by the definition of union. min {x, y} = inf( S1 U S2) by the definition of infimum. Let m = sup(S1) and y = sup(S2) . # m E S1 by the definition of supremum of S1 and n E S2 by the definition of supremum of S2 . m, n E S1 U S2 by the definition of union. max {m, n} = sup(S1 U S2) by the definition of infimum. # SinceS1 U S2 has an infimum and a supremum, it has at least one lower and one upper bound, so it is bounded by the definition of boundedness, infimum, and supremum. Since S1 U S2 is closed and bounded, then by Theorem 3.5.5 (Heine Borel), S1 U S2 is compact