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Unformatted text preview: 2.3 Limits and completeness 35 2.3 Limits and completeness Theorem 2.3.1 (Completeness of R ) . A sequence { x , x 2 ,... } of real numbers converges iff it is a Cauchy sequence. Proof. ( ⇒ ) Assume { x n } converges to x ∈ R . Then fix ε > 0 and find N such that n ≥ N = ⇒  x x n  < ε. Then if j,k ≥ N ,  x j x k  =  x j x + x x k  ≤  x j x  +  x x k  < ε + ε = 2 ε. ( ⇐ ) Assume { x n } is Cauchy. We need to find a Cauchy sequence { y n } ⊆ Q to define y , and then show lim x n = y . 1. For x k , we can find a rational in ( x k 1 k ,x k + 1 k ) by density; call it y k . To see that y k is Cauchy, fix a positive error distance ε > 0 and find N such that j,k ≥ N = ⇒  x j x k  < ε. This is possible, since x n are Cauchy. Then  y j y k  ≤  y j x j  +  x j x k  +  x k y k  < 1 j + ≥ + 1 k . So if we pick j,k so large that 1 j + 1 k < ε , we get  y j y k  < 2 ε , and { y n } is Cauchy....
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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
 Real Numbers, Limits

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