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Sequence proof

  1. Sep 26, 2009 #1
    1. The problem statement, all variables and given/known data

    Let (Tn) be a bounded sequence and let (Sn) be a sequence whose limit = 0. PRove that the limit of (Tn*Sn) = 0. I must complete the proof using only the definition of a limit of a sequence

    2. Relevant equations
    Let Sn be a sequence from N->R with a limit of s
    For all epsilon > 0 there exists an n > N, such that |Sn - s| < epsion
    The suprenum of a sequence is the least upper bound

    3. The attempt at a solution

    I am used to the numerical proofs showing limits I am not comfortable with manipulating these more abstract ones. I cannot solve for N so I am stuck but here is what I am working off of....


    Since Tn is a bounded sequence, by the completeness axiom of R I know that a suprenum exisit and I call it U

    Since Sn is a sequence that converges I know for epsilon > 0 there exist an n > N(For s) such that |Sn|< epsilon

    I need to show that there is an N for any epsilon that implies |Sn*Tn - 0*U| < epsilon

    I was debating proving that the Tn limit is the sup T then provoing the multiplaction of limits rule but that does not sound right there should be a direct method I need some hints on how to start this proof many thanks
     
  2. jcsd
  3. Sep 26, 2009 #2

    HallsofIvy

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    You cannot use that "Tn limit is the sup T then proving the multiplaction of limits rule" because you do not know that Tn has a limit. All you know about Tn is that it is bounded. For example, Tn= (-1)n is bounded but does not converge.

    Since Tn is bounded, let M be an upper bound for |Tn|. Now you know that [itex]-MS_n\le T_nS_n\le MS_n[/itex]. Combine that with the fact that Sn converges to 0.
     
  4. Sep 26, 2009 #3
    Thanks Halls for the response

    Just so I got to close up the proof. I can say for a large N per the hypothesis Sn goes to zero so
    MSn and -MSn go to zero.
    It follows that SnTn goes to zero
    and since epsilon > 0 MSn < epsilon.
    am I done or is it more involved?

    Am I right in saying for these more abstract sequence that you cannot really prove via the definition of a limit of a sequence. The goal is to show that something is less for any given epsilon?
     
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