29.18 Let $f$ be differentialbe on $\mathbb{R}$ with $a = \sup \{|f'(x)|: x \in \mathbb{R} \}<1. $
(a) Select $s_0 \in \mathbb{R}$ and define $s_n = f(s_{n-1}$). Prove $(s_n)$ is a convergent sequence.
Elementary Analysis by Kenneth Ross, 2nd edition p.240
All the proofs of convergence I've seen (not many) start out with "for an $\epsilon > 0$, pick $\dots$". However, to me it felt more natural to put that part at the end, since there was a lot of stuff to develop before I proved that such an $\epsilon$ would work. Is this bad?
I would also appreciate any comments in any of the following areas:
Is it bad to make the proof as conversational as I did?
Is it unorganized or unclear in any way?
Did I appeal too much to the reader's intuition?
Is there a way I could've cut down on the length using this method?
Is there a better method? Thank you.
