k For any given sequence s n s n has a convergent subsequence s n k False Let s

K for any given sequence s n s n has a convergent

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(k) For any given sequence(sn),(sn)has a convergent subsequence(snk).
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(l) For any bounded sequence(sn),lim supsn= sup{sn}. 1 1 1 αis a subsequential limit of a bounded sequence(sn),thenαlim supsn.True:AssumeαSwhereSis the set of subsequential limits of(sn).Then,αsupS.By definition, lim supsn= supS.Therefore,αlim supsn.(n) If every subsequence of a sequence(sn)is convergent, then(sn)itselfmust be convergent.
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(o) If(sn)is a divergent sequence, then some subsequence of(sn)mustdiverge.True:Assume (sn) is a divergent sequence.Assume for the sake ofcontradiction that all subsequences of (sn) converge. Then (sn) itself con-verges (since (sn) is identical to the subsequence (snk) wherenk=k).This is a contradiction, since we assumed (sn) is a divergent sequence.Therefore, some subsequence of(sn)must diverge.(p) If (sn) is unbounded above, then (sn) has a nondecreasing subsequence(snk)which diverges to +.
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2. In each case, prove that the given sequence has an infinite limit.(a)sn=4n2+ 7n-3, wherenN. Provesn+.
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  • Fall '08
  • Staff
  • lim, Mathematical analysis, Limit of a sequence, Limit superior and limit inferior, subsequence

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