Consider $\frac{n}{d}$ and $\frac{n}{c}$. These are also positive divisors of $n$ which satisfy $\frac{n}{d} > \frac{n}{c}$.
We now have
$$\frac{n}{d} - \frac{n}{c} \ge 1,$$
from which
$$c - d \ge \frac{dc}{n} > \frac{d^2}{n}$$
follows.