Could someone please check my work and let me know if my proof is correct

If S is a compact subset of R and T is a closed subset of S , then T' is compact. 1. Prove this using the definition of compactness. Assume S is a compact subset of R and T C S, and T is a closed. Let AT = { An : n E N} be any open cover for T . If Ar is also a cover for S , then since S is compact, by definition 3.5.1 for compactness, there exists a finite subcover of AT, {An _ 1... An - k}, that covers S . Since T C S , then { An _ 1... An _ k} also covers T . Conversely, if AT is not an open cover of S and merely covers T , since S is compact, then by definition 3.5.1 for compactness, there must be an open cover for S . Thus, add an open set to AT in order to cover S. Since T C S , then by the definition of subset, this expanded cover will also cover T . Since T is closed, then by Theorem 3.4.7(b), R - T is open. Add open set, R - T to open set, AT to obtain an open cover of S and thus T. Thus, AT U (IR - T) will cover S and T. Since S is compact and T C S , then definition 3.5.1 for compactness and the definition of subset, # this open cover has a finite subcover , {An _ 1... An _ k} U (R - T) that also covers T . In either case, T is covered by a finite number of open sets. Therefore, T is compact