Chapter 8.1, Problem 52E

(a) Show that if lim_n→∞ a_2n = L and lim_n→∞ a_2n+1 = L, then {a_n} is convergent and lim_n→∞ a_n = L.

(b) If a₁= 1 and

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

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

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