Prove that $\lfloor 2x \rfloor + \lfloor 2y \rfloor \geq \lfloor x \rfloor + \lfloor y \rfloor + \lfloor x+y \rfloor$ for all real $x$ and $y$.

Question

Prove that $\lfloor 2x \rfloor + \lfloor 2y \rfloor \geq \lfloor x \rfloor + \lfloor y \rfloor + \lfloor x+y \rfloor$ for all real $x$ and $y$.

How should I solve this? I can't think of a way with casework and I can't really simplify it more. Thanks in advance for posting a proof!

Answer 1

With these propositions, the trick is usually to write $x=k+e$ and $y=n+f$ where $n$ and $k$ are integers and $0\leq e,g <1$. Then your left side is $2k+\lfloor 2e\rfloor + 2n + \lfloor 2f \rfloor$. And your right side is $k+n+(k+n)+\lfloor e+f \rfloor.$ Cancelling, you need to show now that $\lfloor 2e\rfloor +\lfloor 2f \rfloor \geq \lfloor e+f \rfloor.$ You can do this by considering 4 cases, depending on whether $e$ and $f$ are less than or as big as $1/2$.

E.g. if $0\leq e <1/2$ and $1/2\leq f <1$ then the left side is at least 1 and the right side is at most 1.

Answer 2

It suffices to consider $0\le x\le y<1$. The right hand side is $\le 1$, and it is $>0$ only if $x+y\ge 1$, which requires $y\ge \frac 12$, which makes the left $\ge1$.

Answer 3

In another way, premised that:

$$\begin{gathered} \left\lfloor {x + y} \right\rfloor = \left\lfloor {\left\lfloor x \right\rfloor + \left\{ x \right\} + \left\lfloor y \right\rfloor + \left\{ y \right\}} \right\rfloor = \hfill \\ = \left\lfloor x \right\rfloor + \left\lfloor y \right\rfloor + \left\lfloor {\left\{ x \right\} + \left\{ y \right\}} \right\rfloor = \hfill \\ = \left\lfloor x \right\rfloor + \left\lfloor y \right\rfloor + \left[ {1 \leqslant \left\{ x \right\} + \left\{ y \right\}} \right] \hfill \\ \end{gathered}$$

where the square brackets indicate the Iverson bracket ($\left[ {FALSE} \right] = 0,\;\left[ {TRUE} \right] = 1$)
then

$$\begin{gathered} \left\lfloor {2x} \right\rfloor + \left\lfloor {2y} \right\rfloor = 2\left\lfloor x \right\rfloor + 2\left\lfloor y \right\rfloor + \left[ {1 \leqslant 2\left\{ x \right\} + 2\left\{ y \right\}} \right] \hfill \\ \left\lfloor x \right\rfloor + \left\lfloor y \right\rfloor + \left\lfloor {x + y} \right\rfloor = 2\left\lfloor x \right\rfloor + 2\left\lfloor y \right\rfloor + \left[ {1 \leqslant \left\{ x \right\} + \left\{ y \right\}} \right] \hfill \\ \end{gathered}$$

and, clearly

$$\left[ {1 \leqslant \left\{ x \right\} + \left\{ y \right\}} \right] \leqslant \left[ {1/2 \leqslant \left\{ x \right\} + \left\{ y \right\}} \right]$$