# Thread: Contraction functions

1. ## Contraction functions

A function f from reals to reals is called a contraction of R if and only if there exists a constant r in [0,1) such that x and x' in R we have

|f(x) - f(x')| <_ r|x - x'| ( less than or equal)

Now let f be a contraction of R, with constant r.

a) let x in R be an arbritrary and define a sequence x_n to be f(x_n-1), for each n in the naturals. Show that |x_n+1 - x_n| <_ r^n|x_1 - x_0|.

b) prove X_n is a cauchy sequence.

c)Let p = lim x_n asn n approaches infinity, and prove that f(p) = 0.

in part c) is basically showing every constraction point has a fixed point.

2. a) Let $x_0$ be given, and $f$ have the necessary properties. Then $|x_2-x_1|=|f(x_1)-f(x_0)|\le r|x_1-x_0|$ which satisfies the condition for $n=1$. So assume that the statement is true for some integer $n\ge{1}$. Then
$|x_{n+1}-x_n|=|f(x_n)-f(x_{n-1})|\le r|x_n-x_{n-1}|\le r^n|x_1-x_0|$.

### showing that a function is a contraction mapping,

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