Question 138
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:
$$
\frac{1}{n} \gt \frac{a}{b} \gt \frac{1}{n+1}
$$
Show that when $\frac{a}{b}-\frac{1}{n+1}$ is written in simplest form, its numerator is smaller than $a$.
$$
\frac{1}{n} \gt \frac{a}{b} \gt \frac{1}{n+1}
$$
Show that when $\frac{a}{b}-\frac{1}{n+1}$ is written in simplest form, its numerator is smaller than $a$.