*12. Let D be a nonempty set and suppose that f: D→R and g: D → R. Define the function f +g:D→R by (f +g)(x) = f(x) + g(x). (a) If f(D) and g(D) are bounded above, then prove that (f+ g)(D) is bounded above and sup [(f+g)(D)] S sup f(D) + sup g(D). (b) Find an example to show that a strict inequality in part (a) may occur.
*12. Let D be a nonempty set and suppose that f: D→R and g: D → R. Define the function f +g:D→R by (f +g)(x) = f(x) + g(x). (a) If f(D) and g(D) are bounded above, then prove that (f+ g)(D) is bounded above and sup [(f+g)(D)] S sup f(D) + sup g(D). (b) Find an example to show that a strict inequality in part (a) may occur.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section: Chapter Questions
Problem 63RE
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