Zero is an integer. The point is the right side is not. Since it's strictly greater than zero and less than 1.

Erm, nope, it would be if the sum went to inf but it doesn't.

It is clear that the left hand side has to be zero, however I do not see a straightforward way to prove that without using the contradiction itself and he does not need it to be zero to choose that there is a contradiction, so the claim the left side is an integer sidesteps an entirely optional technical difficulty.

[Edit:] The argument I had in mind is bad, my point is that the value of the left hand side does not matter for the proof, so it is not calculated. (I leave the comment up for context of the answers below.)