A simple floor inequality proof

Question

If $x,y ∈\Bbb R$, I have problems to show that

$$⌊x+y⌋-1 ≤ ⌊x⌋+⌊y⌋ ≤ ⌊x+y⌋$$

Can someone help me?

Answer 1

You know that $\lfloor x\rfloor\le x$ and $\lfloor y\rfloor\le y$; this gives you one of the desired inequalities immediately. To show that $\lfloor x+y\rfloor-1\le\lfloor x\rfloor+\lfloor y\rfloor$, let $m=\lfloor x\rfloor$, $n=\lfloor y\rfloor$, $\alpha=x-m$, and $\beta=y-n$, so that $0\le\alpha,\beta<1$.

• Explain why $\lfloor x+y\rfloor=\lfloor m+n+\alpha+\beta\rfloor=m+n+\lfloor\alpha+\beta\rfloor$.
• Use this to finish the proof.