The proof is outlined as follows: (copied from Wikipedia but Apostol gives the same idea)
If a set is compact, then it is bounded: Consider the open balls centered upon a common point, with any radius. This can cover any set, because all points in the set are some distance away from that point. Any finite subcover of this cover must be bounded, because all balls in the subcover are contained in the largest open ball within that subcover. Therefore, any set covered by this subcover must also be bounded.
My concern is this: the proof seems to implicitly assume that the set is bounded by saying we can cover it with a countable number of concentric open balls. If the set is unbounded, we would not be able to do that since there will be a point in the set that is "infinitely" far from the center of the balls. Or, is my intuition breaking down somewhere?