Unformatted text preview: := lub A . Suppose that Î± 2 n = a . Case 1. If a < Î± 2 , then we observe that Ç« := Î± 2 âˆ’ a 2 is positive. We claim that Î² = Î± âˆ’ Ç« is an upper bound for A , which would contract the statement that Î± is the least upper bound. By the deÂ±nition of Ç« we see that Î² 2 > Î± 2 âˆ’ 2 Ç«Î± > Î± 2 âˆ’ 2 Ç« = a â‰¥ x 2 for each x âˆˆ A . Hence Î² is greater than all x âˆˆ A . Case 2. If Î± 2 < a , then set x = Î± + Ç« where Ç« = min a âˆ’ Î± 2 2 Î± + 1 , 1 . We claim that x âˆˆ A which would contradict that Î± is the least upper bound of A . Indeed, using the deÂ±nition of Ç« , we see that x 2 = Î± 2 + 2 Ç«Î± + Ç« 2 â‰¤ Î± 2 + (2 Î± + 1) Ç« â‰¤ a. â‹„...
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 Fall '10
 Girardi
 Square Roots, positive square roots

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