If S is a compact subset of R and T is a closed subset of S, then T is compact. Prove this using the Heine-Borel theorem. 

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You should remember that Heine-Borel' theorem states that if a subset is bounded and closed, then the subset is compact .

The problem provides the information that S is a compact subset, hence it implies that S is bounded and it is closed.

Since the problem provides the information that T is a subset of S and T is closed and S is closed and bounded, then T is also bounded.

Since a compact set needs to be bounded and closed and since the subset T follows the conditions (bounded and closed), hence, by Heine-Borel's theorem, T is also a compact subset of S.