9. Let be an open cover of and let be the set that contains . Since is a cluster point, also contains all but finitely many . Assume that it covers all but points in .Thus there are at most more open sets in (in addition to ) that completely cover , and so you have proved that every open cover has a finite subcover, and is compact.

I'm not sure about 10. I know in a compact space every sequence has a convergent subsequence, but I don't know how to find two of them.