I need help in each of the problems. See attached picture.
Thanks
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.
For 10 you have to show that there are at least two cluster points in the sequence. Assume that there is only one cluster point
1. First, show that there is such that there are infiniely many 's outside of . If such an does not exist we would have that the sequence converge to
2. For the subsequence formed by the 's outside of , observe that it is in the compact set . Hence there exists a cluster point which can't be .