Defn: Pn converges to point p means that if S is an open interval containing p then there is a positive integer N such that if n is a positive integer and n>N then Pn is in S.

I need to show that if the Pn converges to p and Qn converges to q then Pn+Qn converges to p+q.

I feel like I'm close to getting it, but not quite there yet, so any help would be greatly appreciated.

Note: I can't use the epsilon proof; I must use intervals.