I wanted to ask if this kind of reasoning for proving the result in the title could be considered correct:
We know that: $\left\lceil x\right\rceil =n$ if and only if $n-1<x\leq n$
Then $-\left\lceil x\right\rceil =-n $ if and only if $-n-1<x\leq-n$
Then multiplying by $-1$ the formula $-n-1<x\leq-n$ we get $n+1>-x\geq n$ , inverting the sign.
But $n+1>-x\geq n$ is equivalent to $n\leq-x<n+1$.
We know that $\left\lfloor x\right\rfloor = n$ if and only if $ n\leq x<n+1$.
So from $n\leq-x<n+1$ we can infer that $\left\lfloor -x\right\rfloor =$ $n=-\left\lceil x\right\rceil$