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kingwinner
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Cauchy sequence & "Fixed" point
Suppose that f: Rd->Rd and there is a constant c E (0,1) such that
||f(x)-f(y)|| ≤ c||x-y|| for all x, y E Rd. Let xo E Rd be an arbitrary point in Rd, let xn+1=f(xn). Prove that
a) f is continuous everywhere.
b) (xn) is Cauchy.
c) (xn) 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.
N/A
I proved part a & part b, but I have no idea how to prove parts c & d.
Any help is appreciated!
Homework Statement
Suppose that f: Rd->Rd and there is a constant c E (0,1) such that
||f(x)-f(y)|| ≤ c||x-y|| for all x, y E Rd. Let xo E Rd be an arbitrary point in Rd, let xn+1=f(xn). Prove that
a) f is continuous everywhere.
b) (xn) is Cauchy.
c) (xn) 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.
Homework Equations
N/A
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!