So this question in my book looks like it's essentially asking me to prove the ceiling function exists.
This question is slightly different to other things I found in related question because we're asked to prove the existence of the value in the middle of the inequality.
My initial thoughts are to define an $n \in N$ such that $n > x$ and $-n < x$
Following from this one can say $-n < x \leq n < x + 1$.
Now here I just lose any sort of direction and instead think of trying another method defining a set and using the properties of $inf$ and $sup$ to move from there. The tricky thing I encounter as a roadblock is once again the proving of the existence of the $n$ which is different to many related questions.
Any help or solutions are appreciated, ready and rearing to reply :)