(adsbygoogle = window.adsbygoogle || []).push({}); Cauchy sequence & "Fixed" point

1. The problem statement, all variables and given/known data

Suppose that f: R^{d}->R^{d}and there is a constant c E (0,1) such that

||f(x)-f(y)|| ≤ c||x-y|| for all x, y E R^{d}. Let x_{o}E R^{d}be an arbitrary point in R^{d}, let x_{n+1}=f(x_{n}). Prove that

a) f is continuous everywhere.

b) (x_{n}) is Cauchy.

c) (x_{n}) converges to a limit y.

d) Show that y is a fixed point of f ,that is f(y)=y, and that f has exactly one fixed point.

2. Relevant equations

N/A

3. The attempt at a solution

I proved part a & part b, but I have no idea how to prove parts c & d.

Any help is appreciated!

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# Homework Help: Cauchy sequence & Fixed point

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