On question #1, I was wondering how do you know the epselon ball around x is closed?

*Answer*: Let r=epsilon.

Let S be the set of all y such that d(x,y)=r.

Let T be the complement of S. We must show that T is open.

Let z be a point in T. We must show that there exists q>0 such that the open ball of radius q centered at z is a subset of T.

Case 1: d(x,z) is less than r

Let q=r.

Case 2: d(x,z)>r

Let q=d(x,z)-r. Then use triangle inequality.

*Question*: In case 1 , don't you mean Let q=r-d(x,z)?