7. Under the conditions and notation of Theorem 2 in §1, prove:
(i) When $n \geqslant 1$,
$$\begin{array}{l} Q_{2 n} \geqslant x_{1}\left(x_{2}+x_{4}+\cdots+x_{2 n}\right)+1 \\ Q_{2 n-1} \geqslant x_{1}+x_{3}+\cdots+x_{2 n-1} \\ Q_{n}<\left(1+x_{1}\right)\left(1+x_{2}\right) \cdots\left(1+x_{n}\right) \end{array}$$
(ii) The infinite continued fraction $\left\langle x_{0}, x_{1}, x_{2}, \cdots\right\rangle$ converges if and only if the series $\sum_{i=0}^{\infty} x_{j}$ diverges;
(iii) $\langle 1 / 2,1 / 2,1 / 2, \cdots\rangle=(\sqrt{17}+1) / 4$;
(iv) $\langle 4,1 / 4,4,1 / 4, \cdots\rangle=\sqrt{20}+2$.