Intro to Analysis in-class_Part_12

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

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Unformatted text preview: 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 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....
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This note was uploaded on 11/26/2011 for the course MAT 4944 taught by Professor Wong during the Fall '11 term at FSU.

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

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