hw6solutions - Math 521 - Advanced Calculus I Instructor:...

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Math 521 - Advanced Calculus I Instructor: J. Metcalfe Due: February 3, 2010 Assignment 6 1. 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 X } ≤ sup { f ( x ) : x X } + sup { g ( x ) : x X } and that inf { f ( x ) : x X } + inf { g ( x ) : x X } ≤ inf { f ( x ) + g ( x ) : x X } . Give examples to show that each of these inequalities can be either equalities or strict inequalities. Here we note that for any x , f ( x ) sup f ( x ) , g ( x ) sup g ( x ) . This simply follows as the supremum is an upper bound. Thus, f ( x ) + g ( x ) sup f ( x ) + sup g ( x ) . Hence, sup f ( x ) + sup g ( x ) is an upper bound for { f ( x ) + g ( x ) : x X } . Since the supremum is the least upper bound, it follows that sup x ( f ( x ) + g ( x )) sup x f ( x ) + sup x g ( x ) . The infimums follow similarly. Since the infimum is a lower bound, we have inf f ( x ) f ( x ) , inf
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hw6solutions - Math 521 - Advanced Calculus I Instructor:...

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