Existence of whole number between two real numbers $x$ and $x +1$? How to prove that there is a whole number, integer, between two real numbers $x$ and $x+1$ (in case $x$ is not whole)? 
I need this for an exercise solution in my Topology class, so I can, probably, use more than just axioms from set theory.
Any ideas? 

This can be proved using the decimal expansion of $x$. If $x$ has the decimal expansion $n_0.n_1n_2n_3\cdots$, where $n_0$ is some whole number, then $x+1$ has the decimal expanion $(n_0+1).n_1n_2n_3\cdots$.
Then it is clear that $x < n_0+1 < x+1$ if $x$ is not a whole number itself.
