Prove that given a real number x, there exist unique numbers n and ε such that x=n+ε, where n is an integer, and 0≤ε<1.
I started like this:
x is either an integer or a noninteger. If x=n+ε, then ε=x-n. If x is an integer, then x-n is an integer. So, ε must be an integer. The only integer k such that 0≤k<1 is k=0, so ε=0. This means that n=x-ε=x-0=x.
What do I conclude about the case where x is not an integer?