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Intro to Analysis in-class_Part_14

Intro to Analysis in-class_Part_14 - 2.2 The reals as an...

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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. Then let N = max( N 1 , N 2 ). For m, n N , we have | ( x m + y m ) - ( x n + y n ) | = | ( x m - x n ) + ( y m - y n ) | ≤ | x m - x n | + | y m - y n | < ε + ε = 2 ε.
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