I'm trying to prove the following inequality $\forall x,y \in \mathbb R$ :
$$\lfloor x \rfloor + \lfloor x + y \rfloor + \lfloor y \rfloor \le \lfloor 2x \rfloor + \lfloor 2y \rfloor$$
If one thing is clear for me it is that the basic definition isn't enough to prove this, I would always get an additional $1$ on the left hand side.
Thanks for your trouble.