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## Homework Statement

Show convergence of [itex] s_{n+1}= \sqrt{2+\sqrt{s_n}}[/itex] where [itex]s_1 = \sqrt{2}[/itex]

and that [itex] s_n<2 [/itex] for all n=1,2,3...

## Homework Equations

Let {p_n}be a sequence in metrice space X. {p_n} converges to p iff every neighborhood of p contains p_n for all but a finite number of n.

## The Attempt at a Solution

I'm only assuming that's the relevant property to know...

s_n+1 >=s_n so increasing.

[itex] s_{n+1} > \sqrt{2} [/itex]

so [itex]\frac{1}{s_{n+1}} <\frac{1}{\sqrt{2}}[/itex]

but 1/s_n+1 is positive so

[itex]0< \frac{1}{s_{n+1}} <\frac{1}{\sqrt{2}}[/itex] so it's bounded.

Since it's bounded and increasing, the sequence is convergent.

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