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}

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

Extracted 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="">

Jun 04, 2022

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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}

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