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Let f,g be real-valued functions defined on a nonempty set X satisfying Range f and Range g are bounded subsets of R. Prove each of the following. (a) sup{f(x)+g(x) : x € X}< sup{f(x) : x E X}+ sup{g(x): x € X }. (b) inf{f(x) :x € X}+inf{g(x): x € X} < inf{f(x)+g(x):xEX}. (c) If f(x) < g(x) for all x E X, then sup{f(x) : x € X} < sup{g(x):x € X}.

Let f,g be real-valued functions defined on a nonempty set X satisfying Range f and Range g are bounded subsets of R. Prove each of the following. (a) sup{f(x)+g(x) : x € X}< sup{f(x) : x E X}+ sup{g(x): x € X }. (b) inf{f(x) :x € X}+inf{g(x): x € X} < inf{f(x)+g(x):xEX}. (c) If f(x) < g(x) for all x E X, then sup{f(x) : x € X} < sup{g(x):x € X}.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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7. Let f,g be real-valued functions defined on a nonempty set X satisfying Range f and Range g are bounded subsets of R. Prove each of the following. (a) sup{f(x)+g(x) : x € X}< sup{f(x) : x E X}+ sup{g(x): x € X }. (b) inf{f(x) : x € X}+inf{g(x): x E X }< inf{f(x)+g(x):x€ X}. (c) If f(x) < g(x) for all x € X, then sup{f(x) :x € X} < sup{g(x):x € X }.
Transcribed Image Text:7. Let f,g be real-valued functions defined on a nonempty set X satisfying Range f and Range g are bounded subsets of R. Prove each of the following. (a) sup{f(x)+g(x) : x € X}< sup{f(x) : x E X}+ sup{g(x): x € X }. (b) inf{f(x) : x € X}+inf{g(x): x E X }< inf{f(x)+g(x):x€ X}. (c) If f(x) < g(x) for all x € X, then sup{f(x) :x € X} < sup{g(x):x € X }.
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