Rational Numbers and Finite sets

Question

Let $q$ be an irrational number and fix a positive integer $N$. Consider a small finite interval $(a,b)$ around $q$. Let $U$ be the set of rational numbers of the form $m/n$ such that $gcd(m,n)=1$,$n<N$ and $a<m/n<b$. Show that $U$ is finite.

I know that I must show that there is a bijection from ${1,2,...,m}$ to $U$ but I do not know how to approach this problem.

Answer 1

Hint. Draw a number line on which you mark all the fractions in the unit interval that satisfy the condition when, say $N=6$. Then think about the general case.

You don't need a bijection to prove the set is finite - just an argument saying there can't be more than ...

I think $q$ is irrelevant.

Answer 2

The proposition is false. Consider the rationals of the form $1/n$ where $n$ is allowed to be any negative integer.