a) Let be given, and have the necessary properties. Then which satisfies the condition for . So assume that the statement is true for some integer . Then
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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.