Answer
See below.
Work Step by Step
1. For any real number $x$, let $n\le x\lt n+1$ (where $n$ is an integer), we have $2n\le 2x\lt 2n+2$
2. Since $\lfloor x\rfloor =n$, we get $x-\lfloor x\rfloor =x-n\ge 1/2$ or $x\ge n+1/2$, thus $2x\ge 2n+1$,
3. Combine the above, we have $2n+1\le 2x\lt 2n+2$, thus
$\lfloor 2x\rfloor =2n+1=2\lfloor x\rfloor +1$
