A simple inequality for floor function

Question

If $x,y\in\mathbb R$, I have problems to show that

$$\lfloor x\rfloor+\lfloor y\rfloor\le \lfloor x+y\rfloor\le \lfloor x\rfloor+\lfloor y\rfloor + 1 $$

Can someone help me?

Answer 1

HINT: Let $m=\lfloor x\rfloor$ and $n=\lfloor y\rfloor$, so that $m\le x<m+1$ and $n\le y<n+1$. Then $$m+n\le x+y< m+n+2\;;$$ can you finish it from there?