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JH

# 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 $\lim_{n\to\infty} a_n$ exists.(b) Find $\lim_{n\to\infty} a_n.$

## (a) $$\lim _{n \rightarrow \infty} a_{n}=2$$(b) $$2$$

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