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# Foundations of Analysis MATH 4200

Utah State University

GPA 3.66

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This 2 page Class Notes was uploaded by Benton Yundt on Wednesday October 28, 2015. The Class Notes belongs to MATH 4200 at Utah State University taught by E. Heal in Fall. Since its upload, it has received 20 views. For similar materials see /class/230412/math-4200-utah-state-university in Mathematics (M) at Utah State University.

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Date Created: 10/28/15

Math 4200 Chain Rule Suppose f and g are differentiable functions and w g o f Then 10 is also differentiable and w x g f x Proof Let My 13 if y y yo g yo if y yo It follows that h is continuous at yo Let x W if x 3E 0 f xo if x 10 It follows that t is continuous at 10 wehave MW MW hofz x and so lim h o mo mo g mxon we zazo Math 4200 Theorem Between any two real numbers there exists an irrational number Proof Case 1 Suppose x is positive and x lt 7 There eXists anatural number k such that TZ lt y 7 x Let u TZ Notice that w is an irrational number Starting at x 0 we will walk along the positive real line with a step size equal to w We will prove that we must step into the interval 1 3 since its width is greater than u Let S j jw 2 3 It follows from the Archimedean property that S is not empty Let m be the least element of S Then mu 2 y and m 7 D10 lt y We must show that x lt m 7 D10 Suppose m 7 1w I Then mu 7 u g I and mu 3 x w lt x y 7 x y This implies that mu lt y a contradiction Thus x lt m 7 D10 lt 3 Since 10 is irrational m 7 D10 is also an irrational number Case 2 Suppose x g 0 Choose apositive integer n such that x n gt 0 Consider the positive numbers x n and y n From case 1 there eXists an irrational number u such that x n lt u lt y n Subtracting n from both sides of this inequality gives x lt u 7 n lt 3 Now u 7 n is irrational since u is irrational and n us rational Theorem Between any two real numbers there exists a rational number Proof Modify the proof above by replacing with

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