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## Homework Statement

A sequence [itex]\{s_n\}[/itex] is defined by [itex]s_{n+1} = \frac{1}{2} (s_n + s_{n-1}); s_1 > s_2 > 0[/itex] I have to prove that the sequence is convergent and I have to find the limit.

## Homework Equations

## The Attempt at a Solution

I tried equating the limit of both sides to get s = (1/2)(s + s) but then I just get s = s. I managed to find that the sequence is bounded between [itex]s_1[/itex] and [itex]s_2[/itex] but the sequence is neither increasing nor decreasing.

For proving the convergence I tried used the Cauchy convergence test.

[itex] |\frac{1}{2}(s_n + s_{n-1}) - s_n| = |\frac{1}{2}(s_{n-1} - s_n)| < \in [/itex] is as far as I got. I can't assume that the s_n tends to a limit s because I haven't proved it does yet. So I'm stuck.

I'm also completely stuck on how to prove the convergence of a sequence defined by a recurrence relation in which the relation involves

**TWO**of the previous terms.

where [itex]s_n = f(s_n, s_{n-1})[/itex].