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Solutions for Introduction to Real Analysis | 3rd Edition | ISBN: 9780471321484 | Authors: Robert G. Bartle, Donald R. Sherbert 9780471321484

Solution for problem 11 Chapter 2.4

LetXand Y be nonemptysetsandleth : X x Y --+JRhaveboundedrangeinR Let F: X --+ JR andG

Introduction to Real Analysis | 3rd Edition


Problem 11

LetXand Y be nonemptysetsandleth : X x Y --+JRhaveboundedrangeinR Let F: X --+ JR andG : Y --+ JRbe definedby F(x) := sup{h(x, y) : y e Y}, G(y) := sup{h(x, y) : x eX}. Establish the Principle of the Iterated Suprema: sup{h(x, y) : x eX, y e Y} = sup{F(x) : x eX} = sup{G(y) : y e Y} We sometimes express this in symbols by suph(x, y) = supsuph(x, y) = supsuph(x, y). x,y x y y x 1

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LetXand Y be nonemptysetsandleth : X x Y --+JRhaveboundedrangeinR Let F: X --+ JR andG