HMMT November 2008 T4

Say that $\frac{a}{b}$ is a positive rational number in simplest form, with $a \neq 1$. Further, say that $n$ is an integer such that:

$$\frac{1}{n}>\frac{a}{b}>\frac{1}{n+1}$$

Show that when $\frac{a}{b}-\frac{1}{n+1}$ is written in simplest form, its numerator is smaller than $a$.