# A simple inequality for floor function

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?

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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?
@cth: There’s almost nothing left. Those inequalities leave only two possibilities for $\lfloor x+y\rfloor$. –  Brian M. Scott Mar 13 at 22:38