# Cauchy sequence

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29.18

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

Select s0 in R and define sn = f (sn-1) for n ≥ 1. Thus s1 = f (s0), s2 = f(s1), etc

Prove that (sn) is a convergence sequence. Hint: To show (sn) is Cauchy, first show that |sn+1 - sn| ≤ aּ|sn - sn-1| for n ≥ 1.

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#### Solution Summary

This shows how to prove that a given sequence converges.

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