I am reading the same book and have similar confusions. Plato if you could please elaborate.
To restate what we are trying to show. We are trying to show that if for any convergent sequence in S the limit p is also in S then S will be closed. So we are showing this by contradiction. Supposing that S is not closed...then the complement will not be open and so for a given point p in the complement of S the open ball around p will contain points from S. If I understand correctly at this point we are trying to use the points from S in this open ball around p to construct a sequence that converges to p (which is not in S) which contradicts our hypothesis that the limit of a convergent sequence in S is in S.
The confusion is how do we know that the sequence will converge to p. If n is 100 then the given distance from p is 1/100 and we are asked to choose the p_n's from S from our original ball around p that fall within this distance. So the question is...since we really don't know anything about these p_n's how do we know that as n goes to infinity that there will be p_n's that are within 1/n of p? I.e. that the sequence will converge