Problem on cauchy sequence

Question

Assignment:

Let f be a differentiable on R with a = sup {|f ′(x)|: x in R} < 1.

Select s₀ in R and define s_n = f (s_n-1) for n ≥ 1. Thus s₁ = f (s₀), s₂ = f(s₁), etc
Prove that (s_n) is a convergence sequence.  Hint: To show (s_n) is Cauchy, first show that |s_n+1 - s_n| ≤ a?|s_n - s_n-1| for n ≥ 1.

Provide complete and step by step solution for the question and show calculations and use formulas.