I think I have the idea in my head, but I don't know if I am expressing it clearly in math terms.
The closure of , is defined to be (where are the limit points of )
totally bounded implies that finite number of points s.t
So we already have all the points in S, i.e the union of the balls is a cover for S. We just need to wory about S', the limit points.
so if any limit point is less than away from a point in S. We can say
Right? Is my thinking correct? The limit points are close to the set so we can make sure we get them by making the radius bigger.