Intro to Analysis in-class_Part_15

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

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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. 2. Now show lim x n = y : | y - x k | ≤ | y - y k | + | y k - x k | ≤ | y - y k | + 1 k < ε for k >> 1 . Theorem 2.3.2. Let x n x and y n y . Then 1. x n + y n x + y and x n · y n x · y .

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