Unformatted text preview: Now if A is bounded below then (1) − A = {− x ; x ∈ A } is bounded above. Indeed if b ≤ x for all x ∈ A then − b ≥ − x for all x ∈ A which means − b ≥ y for all y ∈ − A. Now, if sup( − A ) is the least upper bound of − A it follows that − sup( − A ) is a lower bound for A since x ∈ A = ⇒ − x ∈ − A = ⇒ sup( − A ) ≥ − x = ⇒ − sup( − A ) ≤ x. As noted above, if b is any lower bound for A then − b is an upper bound for − A so − b ≥ sup( − A ) and b ≤ − sup( − A ) . This is the deﬁnition of inf A so inf A = − sup( − A ) . 1...
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This note was uploaded on 12/16/2011 for the course STAT 5446 taught by Professor Frade during the Fall '09 term at FSU.
 Fall '09
 Frade

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