3333_10_12_notes.pdf - Section 19 Def Subsequences Given a sequence(sn Let(nk be a sequence of positive integers such that n1 < n2 < n3 < That is(nk is

3333_10_12_notes.pdf - Section 19 Def Subsequences Given a...

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Section 19. Subsequences Def. Given a sequence ( s n ). Let ( n k ) be a sequence of positive integers such that n 1 < n 2 < n 3 < . . . . That is, ( n k ) is an increasing sequence of positive integers. The sequence ( s n k ) is called a subse- quence of ( s n ). NOTE: n k k for all k . 45
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Examples: 1. Let ( s n ) = N = { 1 , 2 , 3 , . . . } ( s n k ) = ( s 2 k - 1 ) = { 1 , 3 , 5 , 7 , . . . } ( s n k ) = ( s 2 k ) = { 2 , 4 , 8 , 16 , . . . } ( s n k ) where n k is the k th prime are subsequences. 46
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2. Let ( s n ) = 1 + ( - 1) n +1 2 = (1 , 0 , 1 , 0 , 1 . . . ) . ( s n k ) = (0 , 0 , 0 , 0 . . . ) e.g., n k = 2 k , or n k = 4 k for all k ( s n k ) = (1 , 1 , 1 , 1 . . . ), e.g., n k = 2 k - 1, ( s n k ) = (1 , 0 , 1 , 0 , 1 . . . ) e.g., n k = k + 2 are subsequences. 47
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THEOREM 1: If ( s n ) converges to s , then every subsequence ( s n k ) of ( s n ) also converges to s . Proof: 48
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Corollary If ( s n ) has a subsequence ( t n ) that converges to α and a subsequence ( u n ) that converges to β with α 6 = β , then ( s n ) does not converge. Proof: 49
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THEOREM 2: Every bounded se- quence has a convergent subsequence. Proof: 50
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THEOREM 3: Every unbounded se- quence has a monotone subsequence that diverges either to + or to -∞ .
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