Since \([x] \le x\) and \([y] \le y\). Adding these two:
$$ [x] + [y] \le x+y $$
This means:
$$ [[x] + [y]] \le [x+y] $$
Since \([[x] + [y]] = [x] + [y]\):
$$ [x] + [y] \le [x+y] $$
Since \([x] \le x\) and \([y] \le y\). Adding these two:
This means:
Since \([[x] + [y]] = [x] + [y]\):