Problem 92E

92. (a) Show that if $\lim _{n \rightarrow \infty} a_{2 n}=L$ and $\lim _{n \rightarrow \infty} a_{2 n+1}=L$, then $\left\{a_{n}\right\}$ is convergent and $\lim _{n \rightarrow \infty} a_{n}=L$.

(b) If $a_{1}=1$ and

$$a_{n+1}=1+\frac{1}{1+a_{n}}$$

find the first eight terms of the sequence $\left\{a_{n}\right\}$. Then use part (a) to show that $\lim _{n \rightarrow \infty} a_{n}=\sqrt{2}$. This gives the continued fraction expansion

$$\sqrt{2}=1+\frac{1}{2+\frac{1}{2+\ldots}}$$