Problem 80E

80. A sequence $\left\{a_{n}\right\}$ is given by $a_{1}=\sqrt{2}, a_{n+1}=\sqrt{2+a_{n}}$.

(a) By induction or otherwise, show that $\left\{a_{n}\right\}$ is increasing and bounded above by $3 .$ Apply the Monotonic Sequence Theorem to show that $\lim _{n \rightarrow \infty} a_{n}$ exists.

(b) Find $\lim _{n \rightarrow \infty} a_{n}$.

Step-by-Step Solution