... but genuinely requires a lot more work to do carefully and properly. People underestimate how much work it is to create a non-handwavey proof of the FTA.

... but genuinely requires a lot more work to do carefully and properly. People underestimate how much work it is to create a non-handwavey proof of the FTA. Tim Gowers has written about this (although I don't have the reference to hand, and I'm on a train with a dodgy connection).