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Intro to Analysis in-class_Part_12

Intro to Analysis in-class_Part_12 - Chapter 2 Construction...

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Chapter 2 Construction of the real numbers The completeness of R . 1 We have seen that Q and R are both ordered fields, so what is the difference? Topology: R is connected, Q is not. For example, 2 is not rational: Q has a “hole” at 2. In topology, a set X is defined to be connected iff there are two nonempty open sets A, B such that A B = X and A B = . Example: Q = ( -∞ , 2) ( 2 , ) . R cannot be written in such a way. Another way to phrase this: completeness . Let A ( b, c ). Then x R such that 1. x is an upper bound for A : a A, a x . 2. y is an upper bound for A = x y . We say x is the least upper bound for A or supremum of A , and write x = sup A . Example: there is no “smallest rational number” that is larger than (or at least as large as) every element of A = (0 , 2). 1 May 2, 2007
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30 Math 413 Construction of the real numbers 2.1 Cauchy sequences Definition 2.1.1. Let { x n } be a sequence in Q . We say the limit of { x n } is L (or that { x n } converges to L ) iff For each m = 1 , 2 , . . . , N m
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