So I am having trouble with this proof, the problem statement is as follows. Suppose $g:[a,b] \rightarrow[a,b]$ is a contraction, and let $\alpha \in (0,1)$ such that for all $x,y \in [a,b]$: $$|g(x)-g(y)| \leq \alpha|x-y|$$Let $x_{n+1}=g(x_{n})$. Prove that if $x^*$ is the fixed point of g, then $|x_n - x^*| < \frac{\alpha^n}{1-\alpha}|x_0-x_1|$. Then it asks to estimate how many iterations it would take.
I think I can handle the second part of the question if I can figure out what $\alpha$ is, but I am just really struggling with the first part of the question. From previous questions that I have done I think that I can use a Taylor series approximation on $g(x)$ to get the left side to the absolute value. If $g'(x^*)\neq 0$ then this approximation would be $$|x_{n+1}-x^*| = |g'(x^*)||x_n-x^*|$$ then I was thinking that the triangle inequality would be useful somewhere else in the proof, but I seem to be having issues, which exactly how useful it would be.