More generally, let $g$ be a positive integer and $M$ a positive real number, and let $G$ be the set of rational numbers $m/n$ such that $m$ and $n$ are positive integers, $m/n\le M$, and $n\le g$; then $G$ is finite.
To see this, note first that there are only $g$ possible denominators for members of $G$, namely, the integers $1,2,\dots,g$. Let $n$ be one of these denominators. Then $m/n\le M$ if and only if $m\le Mn$. In other words, $m/n\in G$ if and only if $m$ is one of the $Mn$ integers $1,2,\dots,Mn$. In particular, since $n\le g$, we must have $Mn\le Mg$. Thus, there are only $g$ possible denominators, and for each of them there are at most $Mg$ possible numerators, so altogether there are at most $g(Mg)=Mg^2$ possible fractions in $G$.