lec14 - Math 117 14 Subsequences Subsequences Denition 14.1 Let(sn be a sequence and let(nk be a strictly increasing sequence of natural n=1 k=1 numbers

lec14 - Math 117 14 Subsequences Subsequences Denition 14.1...

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Math 117 – May 19, 2011 14 Subsequences Subsequences Definition 14.1. Let ( s n ) n =1 be a sequence, and let ( n k ) k =1 be a strictly increasing sequence of natural numbers. Then ( s n k ) k =1 is called a subsequence of ( s n ). For convergent sequences, the subsequences must have the same asymptotic behavior as the entire sequence. Theorem 14.2. If ( s n ) converges to s , then every subsequence of ( s n ) converges to the same limit s . However, even if a sequence does not converge, it may have convergent subsequences. Theorem 14.3. Every bounded sequence has a convergent subsequences. On the other hand, if a sequence is not bounded, it must have subsequences diverging to positive or negative infinities. Theorem 14.4. Let ( s n ) be a sequence of real numbers. ( a ) If ( s n ) is unbounded above, then it has an increasing subsequence diverging to + . ( b ) If ( s n ) is unbounded below, then it has a decreasing subsequence diverging to

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• Fall '08
• Akhmedov,A
• Math, Limit of a sequence, subsequence, lim sup sn, lim inf sn