Could someone please let me know if my proof is correct?

Let K, and Kj be compact subsets of RR. Use The definition of compactness To prove that K, UK, is Compact. Suppose K1 and K2 are compact subsets of R . The definition of By Definition 3.5. of compactness of K] and K2 , since K, and K2 are compact and Ki C K1 U K 2 and K2 C K1 U K2 by the definition of both the union and subset of K1 and K2, then let K1 U K2 C { An: n E N} , which is a family F of open sets that cover K 1 and K2 , or more precisely, K1 U K2 By the same definition, take a subcover of F , G1 = {An _ 1...An _ k}, where G1 C F , then K1 C K1 U K2 C G1 C F . Likewise, take another subcover of F , G2 = {An _ 1... An - j} , where G2 C F , then K2 C K UK2 C G2 CF. Since K1 C G1 and K2 C G2 then by the definitions of the union and subset of G1 and and by G2 , K1 UK2 C GIUG2. The definition of the Additionally, by the definition of the subset of Fand the union of G1 and G2 , since G1 and union of G2 are subsets of F , then G1 U G2 C F. K, and K2 The definition of Since K1 U K2 C G1 U G2 C F, then by definition 3.5.1 of compactness of KI U K2 , K1 U K 2 is compact since it has both a cover F and a finite subcover G1 U G2