The following properties of greatest integer function are given in chapter $4$ of the book An Introduction to the theory of numbers by Niven et al.
The proof for property $(8)$ is given as:
I tried proving $(9)$ as follows:
Let $n$ be the nearest integer to $x$, taking the smaller one if two are equally near. Then $n=x+\theta, -\frac{1}{2}\leq\theta<\frac{1}{2},$ and $-[-x+\frac{1}{2}] = -[-n+\theta+\frac{1}{2}] = n,$ since $0\leq\theta+\frac{1}{2}<1.$
Is the proof correct?
Also, are there any other methods to prove the given properties?