[x] is the integer such that.
It can be shown that this definition is well-defined. Meaning, the greatest integer always exists and is unique. If you wish I can show that to you.
Thus we claim that,
And we can to show that,
Satisfies the same inequality.
Then by well-defineness we have that.
We need to show (2) because (1) is the definition (nothing to prove).
Add "n" to all sides,
And [x]+n is an integer because [x] and "n" are both are and integers are closed under addition.