The defintion of the greatest integer function is.

[x] is the integer such that.

x-1<[x]<=x.

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,

[x+n]=[x]+n.

Meaning,

x+n-1<[x+n]<=x+n.......(1)

And we can to show that,

x+n-1<[x]+n<=x+n.......(2)

Satisfies the same inequality.

Then by well-defineness we have that.

[x+n]=[x]+n

We need to show (2) because (1) is the definition (nothing to prove).

Note that,

x-1<[x]<=x

Add "n" to all sides,

x-1+n<[x]+n<=x+n

And [x]+n is an integer because [x] and "n" are both are and integers are closed under addition.

Thus,

[x]+n=[x+n]