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Unformatted text preview: := lub A . Suppose that 2 n = a . Case 1. If a < 2 , then we observe that := 2 a 2 is positive. We claim that = is an upper bound for A , which would contract the statement that is the least upper bound. By the denition of we see that 2 > 2 2 > 2 2 = a x 2 for each x A . Hence is greater than all x A . Case 2. If 2 < a , then set x = + where = min a 2 2 + 1 , 1 . We claim that x A which would contradict that is the least upper bound of A . Indeed, using the denition of , we see that x 2 = 2 + 2 + 2 2 + (2 + 1) a....
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This note was uploaded on 02/05/2012 for the course MATH 554 taught by Professor Girardi during the Fall '10 term at South Carolina.
- Fall '10
- Square Roots