Let $N>1$ be an integer, which is not a square, and let $p_{k} / q_{k}(k=1,2, \ldots)$ be the convergents to $\sqrt{N}$. Prove that

$$\left|p_{k}^{2}-q_{k}^{2} N\right|<2 \sqrt{N} \quad(k=1,2, \ldots)$$

Explain briefly how this result can be used to generate a factor base $B$, and a set of $B$-numbers which may lead to a factorization of $N$.