Confused with proof exercise real analysis I'm sure it's easy, but I can't see why. Let $p,q,n\in\mathbb{N}$, if $p/q\in(0,1/n)$, then $1/q\in(0,1/n)$.

If $p/q\in(1,1+1/n)$, show that $1/q\in(0,1/n)$.

Attempt: Since $q\in\mathbb{N}$, $0<1/q$. Now, if $p/q\in(1,1+1/n)$, then we can write $p/q$ as 
$$
p/q = 1 + a/b,
$$
where $a/b\in(0,1/n)$, and $a/b$ is rational. By hypothesis, $1/b\in(0,1/n)$. How does one show that $1/q<1/n$? I thought in using contradiction for $1/n\leq 1/q$ but I can't find where everything breaks down.

Since $1 < \frac{p}{q} < 1 + \frac{1}{n}$, we have $0 < \frac{1}{q} \le \frac{p - q}{q} < \frac{1}{n}$.
