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? 

Under the usual Lebesgue measure on $\Bbb R$, the quotient $\Bbb R/\Bbb Z$ has volume 1.
Also the interval $(x,x+1)$ has measure 1 and it is connected.
Therefore the image of $(x,x+1)$ under the quotient map $\pi$ is connected and has measure 1, so it is $\Bbb R/\Bbb Z\setminus\{\bar x\}$. Since $\{\bar x\}\neq\{\bar 0\}$, there must be a point $n\in(x,x+1)$ such that $\pi(n)=\bar 0$. Thus $n\in\Bbb Z$ by definition.
