If x$\displaystyle \notin$ Z (the integers) show that there is exactly one n$\displaystyle \in$Z such that n<x<n+1.
Would I show this by contadiction stating x$\displaystyle \in$Z?
We can suppose that $\displaystyle x>0$ first.
We know that $\displaystyle \left( {\exists n \in \mathbb{Z}} \right)\left[ {x < n} \right]$. By well ordering let $\displaystyle K$ be the least to have that property.
What can be said about $\displaystyle K-1?$
If $\displaystyle x<0$ then do it for $\displaystyle -x$.