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Two monotone sequences proof: Prove lim(an)<=lim(bn)

  1. Apr 14, 2012 #1
    1. The problem statement, all variables and given/known data
    Let (an) be an increasing sequence. Let (bn) be a decreasing sequence and assume that an≤bn for all n. Show that the lim(an)≤lim(bn).


    2. Relevant equations
    I will show that my sequence are bounded above and below, respectively. Thereby forcing the monotone sequence to be convergent, they will have to converge to their respective supremum and infimum.


    3. The attempt at a solution
    Let (an) be an increasing sequence.
    Let (bn) be a decreasing sequence. And let an≤bn for all n.
    Since an≤bn for n, (bn) is bounded below by an for all n.
    Thus since (bn) is monotonically decreasing, there exists an infimum of (bn) s.t. lim(bn)=inf(bn)≥an for all n.
    Similarly lim(an)=sup(an)<bn for all n.
    Since an≤sup(an)≤inf(bn)≤bn for all n,
    lim(an)≤lim(bn).
    Q.E.D.
     
  2. jcsd
  3. Apr 14, 2012 #2

    LCKurtz

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    But you have the bound changing with n. It's easy to fix, but you need to give a single number that is a lower bound for the whole sequence.
     
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