Intro to Analysis in-class_Part_15

Intro to Analysis in-class_Part_15 - 2.3 Limits and...

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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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Intro to Analysis in-class_Part_15 - 2.3 Limits and...

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