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Unformatted text preview: 2.2 The reals as an ordered field 33 2.2 The reals as an ordered field A rigorous argument would construct R as equivalence classes of Cauchy sequences, and then prove: • R is an ordered field. • Every Cauchy sequence in R converges to a point in R . • R satisfies the Axiom of Archimedes. We give the idea. Properties defined on Q can be passed to R by the limit. For example, the field operations: Definition 2.2.1. Let x,y ∈ R . Then pick any rational sequences { x n } , { y n } with lim x n = x and lim y n = y . Define x + y = lim( x n + y n ) and x · y = lim( x n · y n ). This definition only makes sense if { x n + y n } , { x n · y n } are Cauchy: Theorem 2.2.2. If { x n } and { y n } Cauchy in Q , then (i) so is { x n + y n } , and (ii) so is { x n · y n } . Proof. (i) Given ε > 0, we can find N 1 ,N 2 such that m,n ≥ N 1 = ⇒  x m x n  < ε, and m,n ≥ N 2 = ⇒  y m y n  < ε....
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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.
 Fall '11
 Wong

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