given any reel number x there is an integer n such that n>x
this is essentially asking for a proof of the Archimedean Property: If $\displaystyle a>0$ and $\displaystyle b > 0$, then for some integer $\displaystyle n$, we have $\displaystyle na > b$.
i leave it to you to find the proof for this. chances are it is your text, if not (which would be strange) a quick web search should turn up results easily. it is a very famous property.
setting $\displaystyle a = 1$ and $\displaystyle b = x$ you have the claim you want to prove.
minor modifications are needed for the case $\displaystyle x \le 0$