Thread: compact subsets

If S is a compact subset of R and T is a closed subset of S, then T is compact.

A) prove this ussing the definition of compactness
b) prove this using the Heine-Borel theorem

Originally Posted by luckyc1423

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

S is compact thus it is closed and bounded (Heine-Borel Theorem). We only need the fact that S is bounded.

Thus, T subset of S is bounded as well.

But we know T is closed.

Thus, T is a closed bounded subset thus it is compact (Heine-Borel Theorem).