Could someone please let me know if my proof is correct

Prove That if S is a compact subser of IR and T is a closed subset of S, Then T is compact by Heine Bored Theorem. Suppose S is a compact Subser of IR and T is a closed subser of S , Since S is compact, Then by Heine Borel Theorem, S is bounded. Since every subser of a bounded ser is bounded and T is a subser of S. Then T is bounded. Since T is bounded and closed, Then by Heine Borel Theorem, T is compact