Question

Prove that every closed interval in r is compact

Answer 1

If you're looking for a proof that doesn't use Heine-Borel, here are some hints:

Notice that [a,b] can be rewritten as [a,a+b2]∪[a+b2,b].Use contradiction, i.e, assume that some open cover of [a,b] has no finite subcover and wlog no finite subcollection covers [a,a+b2].Keep goingYou will need the nested intervals theorem.