Math Help - Recursion Problem!

1. Recursion Problem!

Let x_0 be a real number in the interval (0,1] and {x_n} be sequence for all n >= 0, defined inductively by setting x_n = sin(x_(n-1)) for all n >= 1. I have already proved that {x_n} is a strictly decreasing sequence in (0,1] and that the lim x_n asn→∞ = 0.

We need to prove parts 2 and 3 in the attachment!

Thanks!

3. Re: Recursion Problem!

Hi,
A solution to the 2nd problem is just a lengthy application of L'hopital's rule. The attachment has most of the details:

4. Re: Recursion Problem!

Hi again,
A "simpler" solution than my previous post is provided by the attachment. At least the derivatives involved are easily calculated.