sup{f(x) + g(x): x e X) < sup(f (x) : x e X} + sup(g(x): x e X) inf{f(x) : x e X} + inf{g(x): x e X) < ipf{f(x) +g(x) : x e X).
sup{f(x) + g(x): x e X) < sup(f (x) : x e X} + sup(g(x): x e X) inf{f(x) : x e X} + inf{g(x): x e X) < ipf{f(x) +g(x) : x e X).
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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Transcribed Image Text:7. Let X be a nonempty set, and let f and g be defined on X and have bounded ranges in R. Show
that
sup{f(x) + g(x): x e X} < sup{f (x) : x e X} + sup{g(x) : x e X}
and that
inf{f(x): x € X} + inf{g(x) : x e X} < ipf{f(x) +g(x) : x e X}.
Give examples to show that each of these inequalities can be either equalities or strict inequalities.
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