http://web.williams.edu/Mathematics/lg5/harmonic.pdf seems to me, this proof relies on the principle of the excluded middle, while the term-grouping proof does not. So the latter is better. Intuitionism, baby. I've always felt it would be a somehow desirable thing to go through mathematics determining which parts were provable without relying on excluded middle. That project has never really been done. On the other hand, maybe I'm just nuts.