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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 > 0 and find N such that n N =  x x n  < . Then if j,k N ,  x j x k  =  x j x + x x k   x j x  +  x x k  < + = 2 . ( ) 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 > 0 and find N such that j,k N =  x j x k  < . 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 < , we get  y j y k  < 2 , and { y n } is Cauchy....
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 Fall '11
 Wong
 Real Numbers, Limits

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