On this, am I trying to prove that X is complete by showing that every Cauchy sequence converges to some point in X? has nothing to do with #1.

Ifis a finite subset of a metric spaceFjust show that the complement ofXis an open set. For any pointFnot intit is easy to construct an ball centered atFcontaining no other point oft.F

For #2, you know that every irrational number is the limit of a sequence of rational numbers. Put another way:Every irrational number is a limit point of the set of rational numbers.