Real An 2. Assume that f : R → R is such that |f(x) – f(y)|< \/x – y| for all x, y E Rand some A E (0, 1).(a) Prove that for every r > 0 one has thatf (r) – r < f(0) – (1 – A)r and f(-r)+r > f(0) + (1 – A)r(b) Consider g(x) = f(x) –g(r*) < 0 and g(-r*) > 0.x. Prove that there exists r* > 0 such that(c) Assume that r* is the number from part (b). Prove that there existsx* € (-r*, r*) such that g(x*) = 0 or, equivalently, f(x*) = x*.

Explanation Given - assume that f:ℝ→ℝ such that f(x)-f(y)≤λx-y for all x, y ∈ℝ and some λ∈(0, 1) To…

Assume that f : R → R is such that |f(x) – f(y)| < \|x – y| for all x, y E R and some A E (0, 1). (a) Prove that for every r > 0 one has that f(r) – r< f(0) – (1 – A)r and f(-r)+r> f(0) + (1 – A)r (b) Consider g(x) g(r*) < 0 and g(-r*) > 0. (c) Assume that r* is the number from part (b). Prove that there exists x* € (-r*, r*) such that g(x*) = 0 or, equivalently, f(x*)= x*. f (x) – x. Prove that there exists r* > 0 such that

Assume that f : R → R is such that |f(x) – f(y)| < \|x – y| for all x, y E R and some A E (0, 1). (a) Prove that for every r > 0 one has that f(r) – r< f(0) – (1 – A)r and f(-r)+r> f(0) + (1 – A)r (b) Consider g(x) g(r) < 0 and g(-r) > 0. (c) Assume that r* is the number from part (b). Prove that there exists x* € (-r, r) such that g(x) = 0 or, equivalently, f(x)= x. f (x) – x. Prove that there exists r > 0 such that

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.4: Ordered Integral Domains

Problem 1E: Complete the proof of Theorem 5.30 by providing the following statements, where and are arbitrary...

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