Prove that $-\lfloor x+1 \rfloor = \lfloor -x \rfloor$ for all real $x$ not an integer

Question

Let $\lfloor x \rfloor$ be the greatest integer lower bound of any real number $x$. Prove that if a real $x$ is not an integer, then $$-\lfloor x+1 \rfloor = \lfloor -x \rfloor.$$

Answer 1

Let $n=\lfloor x\rfloor$ then we have $$n<x<n+1$$ so

$$n+1<x+1<n+2\implies \lfloor x+1\rfloor=n+1$$ and

$$-n>-x>-n-1\implies \lfloor -x\rfloor =-n-1$$ and the result follows.