# HW#3 - T it is also a lower bound of S But inf S is the...

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SOLUTIONS TO HOMEWORK # 3 4.5 sup S is an upper bound of S , hence for each s S we have s sup S . Thus, if sup S S , it is the largest number in S , i.e. it coincides with max S . 4.6 (a) Let s S . Since inf S is a lower bound of S and sup S is an upper bound of S , we have inf S s and s sup S . It remains to use the transitivity of the order. 4.6 (b) For each s S we have inf S s sup S ; inf S = sup S implies s = inf S = sup S so s is the unique element in S . 4.7 (a) From 4.6(a) we know that inf S sup S . Since S T and sup T is an upper bound of T , it is also an upper bound of S . But sup S is the least upper bound of S and hence sup S sup T . Similarly, since inf T is a lower bound of
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Unformatted text preview: T , it is also a lower bound of S . But inf S is the greatest lower bound of S and hence inf T ≤ inf S . 4.13 The equivalence of (a) and (b) follows from Problem 3.7(b). (c)is simply a notation for (b). Remark . The relation sup S ∈ S is a special assumption in Problem 4.5, and it says that namely in this special case max S exists and sup S = max S . Be aware: this is not always true! Counterexample: if S = (0 , 1) then inf S = 0 , sup S = 1 and both are not contained in S ! And S has neither min S , nor max S ! 1...
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