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Post Made Community Wiki by Martin Brandenburg
1

Real number happens to be an integer

Sometimes you have a real number (with a rather complicated definition), and with some effort you can show that

1) this real number is, actually, an integer;

2) the distance of this real number to an integer, say $0$, is less than $1/2$.

Thus you can conclude that this real number is $0$! I think this is a very nice trick. Especially when the argument for 1) is so involved that you don't really see this a priori. However, I don't remember in which context I have seen this. But I guess that this trick works in various situations.

So my question is: Can you give nice, explicit instances of this trick?