Chapter 11.1, Problem 92E

(a) Show that if ${{\displaystyle \lim }}_{n\to \infty }{a}_{2n}=L$ and ${{\displaystyle \lim }}_{n\to \infty }{a}_{2n+1}=L$, then $\left\{a_n\right\}$ is convergent and ${{\displaystyle \lim }}_{n\to \infty }{a}_n=L$.

(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 ${{\displaystyle \lim }}_{n\to \infty }{a}_n=\sqrt{2}$. This gives the continued fraction expansion

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