Real analysis Please prove with these steps. Function contraction theoremLet (X, d) be a complete metric space.Suppose that f: XX satisfiesd(f(x), f(y)) ≤ 2-¹d(x, y) for all x, y E X.Show that there exists a unique point x* EX such that f(x*) = x*.Problem 5 Hint:i=0Step 1. Let xo E X. Define a sequence {x} by the recurrence relationxi = f(xi-1) for i=1,2,....Step 2. Show that {x}Step 3. Show that x* is the point of interest.converges to a point x* by showing that it is Cauchy.

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Function contraction theorem Let (X, d) be a complete metric space. Suppose that f: XX satisfies d(f(x), f(y)) ≤ 2-¹d(x, y) for all x, y ≤ X. Show that there exists a unique point x* EX such that f(x*) = x*.

Function contraction theorem Let (X, d) be a complete metric space. Suppose that f: XX satisfies d(f(x), f(y)) ≤ 2-¹d(x, y) for all x, y ≤ X. Show that there exists a unique point x* EX such that f(x) = x.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.2: Mappings

Problem 27E: 27. Let , where and are nonempty. Prove that has the property that for every subset of if and...

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Question

Real analysis

Please prove with these steps.

Function contraction theorem
Let (X, d) be a complete metric space.
Suppose that f: XX satisfies
d(f(x), f(y)) ≤ 2-¹d(x, y) for all x, y E X.
Show that there exists a unique point x* EX such that f(x*) = x*.
Problem 5 Hint:
i=0
Step 1. Let xo E X. Define a sequence {x} by the recurrence relation
xi = f(xi-1) for i=1,2,....
Step 2. Show that {x}
Step 3. Show that x* is the point of interest.
converges to a point x* by showing that it is Cauchy.

Branch of mathematical analysis that studies real numbers, sequences, and series of real numbers and real functions. The concepts of real analysis underpin calculus and its application to it. It also includes limits, convergence, continuity, and measure theory.

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