In the next two questions, X is a nonempty set and f: X R is...

 

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In the next two questions, X is a nonempty set and f: X → R is a function. For any ACX, we denote supAf = sup{f(x) | x A). Also, for any functions f.g: X → R we denote f + g: X → R the function defined by (f+g)(x) = f(x) + g(x) for all * EX and cf: X → R the function defined by (cf)(x) = cf(x) for all x € R. Show that (i) sup (cf) = csup (f) and inf(cf) = cinf (f), provided that c> 0. (ii) sup{f(x) - g(y) | xe A, ye A} = supA(f) - infA(g). In the next two questions, X is a nonempty set and f: X → R is a function. For any ACX, we denote supAf = sup{f(x) | x A). Also, for any functions f.g: X → R we denote f + g: X → R the function defined by (f+g)(x) = f(x) + g(x) for all * EX and cf: X → R the function defined by (cf)(x) = cf(x) for all x € R. Show that (i) sup (cf) = csup (f) and inf(cf) = cinf (f), provided that c> 0. (ii) sup{f(x) - g(y) | xe A, ye A} = supA(f) - infA(g).

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Step 14 Given f X R ACX supAf supfxx A To Find i supA cf c supAf and infA cf c inff ii sup fx gy x A ... View the full answer