By definition, \([x+n]\) is the greatest integer less than or equal to \(x+n\):
$$ [x+n] \le x+n \lt [x+n]+1 $$
By definition, we also know:
$$ [x] \le x \lt [x]+1 $$
We can add \(n\):
$$ [x] + n \le x + n \lt [x]+n+1 $$
This shows that \([x]+n\) is also the greatest integer less than or equal to \(x+n\).
If both \([x]+n\) and \([x+n]\) are the greatest integers below \(x+n\), then \([x]+n = [x+n]\).