Here is my understanding: let $x,y$ be real.
W.L.O.G, $x<y$; then consider the difference between $x$ and $y$, $y-x$. If the difference is greater than 1 then we are done because there is an integer between them. Otherwise, multiply them by some large enough integer $n$ so that their difference after multiplication is greater than 1. Then there is an integer $m$ between them: $nx<m<ny$. from that we divide all side of inequality by n to conclude the proof!