3. Recall that a rational number r EQ is a number of the form r = where a, b & Z, b +0. Decide whether each of the following claims is true or false. Then prove your answers. Claim 1. There is a positive rational number smaller than any other positive rational number. Claim 2. For any r € Q with r = 0, there exists m € Z for which rm € N = {1,2,3, ...}. 4. Suppose that r € Q, r‡ 0, and that x € R\Q (x is irrational). Prove that rx is also irrational. 5. Prove that for any two nonnegative real numbers, their arithmetic mean is always greater than or equal to their geometric mean. Meaning, prove that for any x,y ER where x, y ≥ 0, x+y 2 Σvry. Moreover, prove that there is equality if and only if x = y.

3. Recall that a rational number r EQ is a number of the form r = where a, b & Z, b +0. Decide whether each of the following claims is true or false. Then prove your answers. Claim 1. There is a positive rational number smaller than any other positive rational number. Claim 2. For any r € Q with r = 0, there exists m € Z for which rm € N = {1,2,3, ...}. 4. Suppose that r € Q, r‡ 0, and that x € R\Q (x is irrational). Prove that rx is also irrational. 5. Prove that for any two nonnegative real numbers, their arithmetic mean is always greater than or equal to their geometric mean. Meaning, prove that for any x,y ER where x, y ≥ 0, x+y 2 Σvry. Moreover, prove that there is equality if and only if x = y.

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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