I guess what the professor was getting at is how do you know the p-series converges by virtue of p>1? I just sat down and ran through my thinking and saw that eventually the terms add nothing to the end, but I didn't prove anything. In order to mathematically KNOW that it converges, is it enough to know that all p-series with p>1 converge or do you have to go further and be able to demonstrate for any given p-series p>1, it does converge?