Math 117 – May 19, 201114SubsequencesSubsequencesDefinition 14.1.Let (sn)∞n=1be a sequence, and let (nk)∞k=1be a strictly increasing sequence of naturalnumbers. Then (snk)∞k=1is called a subsequence of (sn).For convergent sequences, the subsequences must have the same asymptotic behavior as the entiresequence.Theorem 14.2.If(sn)converges tos, then every subsequence of(sn)converges to the same limits.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 ornegative infinities.Theorem 14.4.Let(sn)be a sequence of real numbers.(a) If(sn)is unbounded above, then it has an increasing subsequence diverging to+∞.(b) If(sn)is unbounded below, then it has a decreasing subsequence diverging to
You've reached the end of your free preview.
Want to read the whole page?
Math,Limit of a sequence,subsequence,lim sup sn,lim inf sn