add-mult-are-cts - Taking absolute-values, then...

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Multiplication in C is continuous Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA squash@math.ufl.edu Webpage http://www.math.ufl.edu/ squash/ 8 November, 2008 (at 23:18 ) Abbreviations. Use posreal for “positive real number”. A sequence ~ x abbreviates ( x 1 , x 2 , x 3 , . . . ). Use Tail N ( ~ x ) for the subsequence ( x N , x N + 1 , x N + 2 , . . . ) of ~ x . 1 : Addition-Cts theorem. The addition operation C × C C is continuous. Restated: Suppose ~ x ,~ y C with lim( ~ x ) = α and lim( ~ y ) = β . With p n : = x n + y n , then, lim( ~ p ) = α + β . Proof. Fix a posreal ε . Take N large enough that Tail N ( ~ x ) Bal ε 2 ( α ) and Tail N ( ~ y ) Bal ε 2 ( β ) . Each index k has p k - [ α + β ] = [ x k - α ] + [ y k - β ]. For k > N , then, ± ± ± p k - [ α + β ] ± ± ± 6 | x k - α | + | y k - β | 6 ε 2 + ε 2 = ε . ± Remark. The same theorem and proof hold for addition on a normed-vectorspace; simply replace |·| by the norm k·k . ² Abbreviations.
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This note was uploaded on 01/26/2012 for the course MTG 6401 taught by Professor Staff during the Fall '09 term at University of Florida.

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