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math-2011-note5 - 9 Limits and Open Sets(ch.12 9.1...

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9 Limits and Open Sets (ch.12) 9.1 Sequences in Real Numbers A sequence of real numbers { x 1 , x 2 , ··· , x n , ···} is an assignment of a real number x n to each natural number n . Definition 7 (p.255) . Let { x 1 , x 2 , ··· , x n , ···} be a sequence of real numbers. A real number r is the limit of this sequence if for any positive number ε , there is an integer N such that | x n - r | < ε for all n N . In this case we say that the sequence (or x n ) converges to r and write lim n x n = r , lim x n = r , or x n r . Theorem 19 (Thm 12.1, p.256) . A sequence can have at most one limit. Theorem 20 (Thm 12.2, 12.3, 12.4, p.257-9) . Let { x n } n = 1 and { y n } n = 1 be sequences with limits x and y, respectively. Then the sequence { x n + y n } n = 1 converges to the limit x + y and the sequence { x n y n } n = 1 converges to the limit xy . If x n ( ) b for all n, then x ( ) b . Definition 8 (p.256) . Given a sequence { x 1 , x 2 , ··· , x n , ···} and an infinite set { n 1 , n 2 , ··· , n j , ···} ⊂ { 1 , 2 , 3 , ···} such that n 1 < n 2 < ··· < n j < ··· , a sequence { x n 1 , x n 2 , ··· , x n j , ···} is called a subsequence of the original sequence. 9.2 Sequences in R m A sequence in R m , { x 1 , x 2 , ··· , x n , ···} is an assignment of a vector x n in R m to each natural number n . Given two vectors x and y in R m , the distance d ( x , y ) between them is d ( x , y ) = bardbl x - y bardbl = radicalbig m i = 1 ( x i - y i ) 2 .
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