Does anyone have an early citation for this proof, or know who might have thought it up? Byers doesn't give any citation for the idea in his book.
Dear Joshua, years ago I came up with the same proof and asked in a math forum pretty much the same questions you did. Nobody could name an early source but they confirmed that the proof works. Hopefully some other reader can shed some light on the question. (I conjecture that the proof has been thought of hundreds of times, finding an early citation is another thing) Another "classical" proof would be: If the sum converges, then, for some N, we have 1/N + 1/(N+1) + ... < 1/2. On the other hand, if we take the first N+1 terms of the series 1/N + 1/(N+1) + ... + 1/(2*N) > (N+1)/(2*N) > 1/2 (because we have (N+1) terms and all of them are bigger than 1/(2N)). Contradiction. Best wishes, Stefan