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.

Answered: Image /qna-images/answer/7a787b83-6d75-4910-a1a8-d54c4f3d1be6.jpg

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.

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter3: The Derivative

Section3.CR: Chapter 3 Review

Problem 4CR

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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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