Chapter 4 - Elementary Number Theory and Methods of Proof - Exercise Set 4.5 - Page 197: 25

Let $x$ be an arbitrary real number. Then $\lfloor x/2\rfloor=m$ for some integer $m$ such that $m\leq x/2$

Note that, when we divide the proof into cases, each "sub-proof" is independent of the other. Hence, we cannot use results derived in one to draw a conclusion in the other.