PROBLEM 5. Let S be a nonempty subsets of R. Suppose f: S R and g:...
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PROBLEM 5. Let S be a nonempty subsets of R. Suppose f: S→ R and g: S→ R are bounded functions (this means their ranges are bounded subsets of R. (a) Show that {f(x) + g(x): x € S} c{f(x): x € S} + {g(x) : x = S} (b) Prove that sup{f(x) + g(x) : x = S} ≤ sup{f(x) : x € S} + sup{g(x) : x = S}. (c) Give an example of two functions for which the inclusion in part (a) is strict. (d) Show that the inequality in part (b) can be strict. PROBLEM 5. Let S be a nonempty subsets of R. Suppose f: S→ R and g: S→ R are bounded functions (this means their ranges are bounded subsets of R. (a) Show that {f(x) + g(x): x € S} c{f(x): x € S} + {g(x) : x = S} (b) Prove that sup{f(x) + g(x) : x = S} ≤ sup{f(x) : x € S} + sup{g(x) : x = S}. (c) Give an example of two functions for which the inclusion in part (a) is strict. (d) Show that the inequality in part (b) can be strict.
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Related Book For
Introduction to Real Analysis
ISBN: 978-0471433316
4th edition
Authors: Robert G. Bartle, Donald R. Sherbert
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