Show that two sets are disjoint I'm trying to solve this problem about two different sets. I have to show that they are disjoint. I understand that they need to be disjoint, because they are two different real ranks, but I dont know how to prove it.
Thank you so much for the help. Any hint is welcome!

Show that if two differentintegers $m\neq n$ the following sets are disjoint: $(n-1,n]=\{x\in \Re : n-1<x\leq n \}$ and $(m-1,m]=\{x\in \Re : m-1<x\leq m \}$


Without loss of generality we can assume $n>m$ (the case $n<m$ follows by symmetry). Then observe that $m-1<m\leq n-1<n$.
We proceed using proof by contradiction. Suppose that there is some $x\in(m-1,m]\cap(n-1,n]$. Then $m-1<x\leq m$ and $n-1<x\leq n$. We have $x>n-1\geq m$ and $x\leq m$, absurd!
