Show that $\inf \{ m+n\omega: m+n\omega>0,and,m,n\in{Z}\}= 0$, where $\omega>0 $ is irrational. Let $\omega\in\mathbb {R}$ be an irrational positive number. Set
$$A=\{m+n\omega: m+n\omega>0,and,m,n\in{Z}\}.$$
Show that $\inf{A}=0.$
How should I start this problem? I don't get this problem.

Fix $\epsilon>0$. Fix an integer $N>1/\epsilon$. Consider the fractional parts  of the numbers $k\omega, 0<k\le N$. These are commonly denoted
$$
\{k\omega\}:=k\omega-\lfloor k\omega\rfloor.
$$
Because $\omega$ is irrational the numbers $\{k\omega\}\in(0,1)$, $k=1,2,\ldots,N$, are all distinct.
Because there are $N$ of them some two of them, say $\{k\omega\}$ and $\{\ell\omega\}$ are within $\epsilon$ of each other (you may use the pigeonhole principle to see this). 
If $k>\ell$ (resp. if $k<\ell$) then $0<\{t\omega\}<\epsilon$ for $t=k-\ell$ (resp. $t=\ell-k$). This means that
$$0<n\omega+m<\epsilon$$
for $n=t$ and $m=-\lfloor t\omega\rfloor$.

Making this CW for this is surely a duplicate, but I don't have time to look for one.
