7. Let $u_{0}>u_{1}>1$ as in Theorem 4 of § 3; and let $b_{0}=1, b_{1}=2$ and
$$b_{j+1}=b_{j}+b_{j-1}, \quad j=1,2, \cdots$$

Then, in the notation of Theorem 4 of § 3, we have $u_{1} \geqslant b_{k}$. Furthermore, prove that:
$$k+1 \leqslant 2\left(\ln u_{1}\right) / \ln 2$$

Explain the significance of this result.