If $1\le q\le n$ and $1\le kq\le n$, then $\left|\frac{p}{q}-\frac{p'}{kq}\right|\ge\frac{1}{kq}\ge\frac{1}{n}.$ However, we cannot have two such fractions in an open interval of length $\frac{1}{n}$. Clearly we cannot have two fractions with equal denominator, so if $n=2m$, there are at most $m$ fractions, and if $n=2m+1$, there are at most $m+1$ fractions, so we are done.