Proof $ \lfloor{x}\rfloor + \lfloor{y}\rfloor \le \lfloor{x + y}\rfloor $ I want to prove that $ \lfloor{x}\rfloor + \lfloor{y}\rfloor \le \lfloor{x + y}\rfloor $, I started proving but I got stuck.
Let $ x, y \in \mathbb{R} $. Therefore:
$ \lfloor{x}\rfloor \le x $
$ \lfloor{y}\rfloor \le y $
$ \Rightarrow $
$ \lfloor{x}\rfloor + \lfloor{y}\rfloor \le x + y $

Is this proof OK, using both (1) integers $n = \left\lfloor n \right\rfloor$, and (2) if $a\le b$ then $\left\lfloor a\right\rfloor \le \left\lfloor b \right\rfloor$?
$$\left\lfloor x\right\rfloor + \left\lfloor y \right\rfloor
= \big\lfloor \left\lfloor x\right\rfloor + \left\lfloor y \right\rfloor \big\rfloor
\le \left\lfloor x + y \right\rfloor$$
