SquareRoot - := lub A . Suppose that 2 n = a . Case 1. If a...

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Math 554- 703 I - Analysis I Existence of Square Roots Theorem. If a is a nonnegative real number, then there exists a unique positive real number α such that α 2 = a . We use the notation a := α. Lemma. Positive square roots are unique. Proof. Suppose not. If x < y and x, y are both positive square roots of a > 0, then x 2 < xy < y 2 . But x 2 = a = y 2 . Contradiction. Proof of the Theorem. First notice that we may assume without loss of generality that 0 < a < 1. If a = 1, then α = 1 is the unique square root of a . If 1 < a , then b := 1 /a is less than 1, and we denote its square root by β . We set α := 1 , then α 2 = 1 2 = 1 /b = a . Also notice that in the case 0 < a < 1, the Lemma and its proof imply that 0 < α < 1. For 0 < a < 1, we de±ne the set A := b x > 0 | x 2 a B . Notice that A is nonempty ( a A ) and bounded from above by 1, so let α
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Unformatted text preview: := lub A . Suppose that 2 n = a . Case 1. If a &lt; 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 &gt; 2 2 &gt; 2 2 = a x 2 for each x A . Hence is greater than all x A . Case 2. If 2 &lt; 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.

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