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Unformatted text preview: 2. By induction, the distance between g n ( x ) and √ 2 is less than 1 2 n of the distance between x and √ 2. Thus, in this case as well, g n ( x ) → √ 2 as n → ∞ . The last case to consider is x < √ 2. In this case for any c between x and √ 2 we must have g ( c ) < 0. Using the Mean Value Theorem again, this implies that g ( x ) would be greater than √ 2. Thus, all the elements of the sequence { g n ( x ) } ∞ n =1 are greater than √ 2 and the previous case applies to show that g n ( x ) → √ 2 as n → ∞ . A similar argument shows that for any x < 0, g n ( x ) → √ 2 as n → ∞ . 2...
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 Summer '08
 Kyung
 Logic, Statistics, Mean Value Theorem, JAMES KEESLING

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