Could someone please check my work

Prove That if S is a compact subser of FR and T is a closed subset of S, Then I is compact by Heine - Bore) Theorem. Assume Tis a closed subset of S. Assume S is a compact subset of MR. Then by The Heine-Barel Theorem, S is bounded. Since T is a subset of S Then I is also bounded by The bounds of S. Since I is close and bounded Then by The Heine-Barel Theorem, T is compact