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?