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Intro to Analysis in-class_Part_19

# Intro to Analysis in-class_Part_19 - 3.1 Limits and bounds...

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3.1 Limits and bounds 43 Example 3.1.4. The sequence { ( - 1) n (1+ 1 n ) } has the monotone decreasing subsequence 1 + 1 2 n obtained by taking every second term. Infinitely many deletions. x n = - 2 , 3 2 , - 4 3 , 5 4 , - 6 5 , 7 6 , . . . n = 1 , 2 , 3 , 4 , 5 , 6 , . . . n 1 = 2 , n 2 = 4 , n 3 = 6 , . . . x n 1 = x 2 = 3 2 x n 2 = x 4 = 5 4 , x n 3 = x 6 = 7 6 , Note: n k k . Example 3.1.5. The sequence { 1 , 1 2 , 1 3 , 1 4 , . . . } has subsequence { 1 2 , 1 3 , 1 4 , . . . } obtained by deleting the first term. A single deletion. Theorem 3.1.13. Let { x n } be a sequence in R . (i) x n x iff every neighbourhood of x of the form ( x - ε, x + ε ) , ε > 0 contains all but finitely many points x n . (ii) x R is a limit point of the set A iff there is a sequence { x n } ⊆ A with x n x and x n 6 = x . (iii) x n x iff x n k x , for every subsequence { x n k } . (i). ( ) Suppose x n x and fix any ε > 0. Corresponding to this ε , there is an N N such that n N = ⇒ | x n - x | < ε, by defn of convergence. Thus, all points save { x 1 , . . . , x N - 1 } must lie in the interval.

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Intro to Analysis in-class_Part_19 - 3.1 Limits and bounds...

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