If S is a compact subset of R and T is a closed subset of S,then T is compact.
(a) Prove this using definition of compactness.
(b) Prove this using the Heine-Borel theorem.
My solution: firstly I should suppose a open cover of T, and I still need to think of the set S-T. But if S-T is open in R,it can be done because the open cover of T and S-T is a open cover of R. The reality is S-T is not necessarily a open set in R. My question is that How we can find a open cover which covers S-T but misses T! I don't know how to do this thing!
in terms of part (b), I know it is bounded, but How to prove it is closed for T.