On page 7 of the book 'An Introduction to Numerical Analysis' by Suli and Mayers a proof of the Contraction Mapping Theorem is given. Hopefully somebody has the book or can understand the equations I'm writing here as is because I have left out alot of the background info (as the proof draws on many previous statements).
While proving that $(x_k)$ converges to the fixed point $\epsilon$ the book states the following -
1) $|x_k - \epsilon| = |g(x_{k-1}) - g(\epsilon)| \leq L|x_{k-1} -\epsilon|$ with $k \geq 1$
It then says 'we can deduce by induction' that
2) $|x_k - \epsilon| \leq L^k|x_{k0} -\epsilon|$ with $k \geq 1$
I can't see how to show by induction that 1) leads to 2). I can see the trivial base case where $k=1$ but I can't see how the inductive step is handled. Any ideas?