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Math Help - Sequence proof

  1. #1
    Senior Member Danneedshelp's Avatar
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    Sequence proof

    Q: Consider a sequence (I_{n})_{n} of closed nested intervals, I_{n} = [a_{n} , b_{n} ]. Assume also that
    the sequence of the lengths of these intervals, (l_{n})_{n} with l_{n}=b_{n}-a_{n} , converges to zero. Show that the
    intersection \bigcap_{n}\\I_{n} consists of just one point.

    A: My thinking is to use the axiom of completeness to conclude (I_{n}) is bounded. Then, by the nested interval property we know the intersection is non-void and that we have a monotone sequence. And, since (I_{n}) is bounded and monotone, by the monotone convergence theorem we know (I_{n}) is convergent. We know that (I_{n}) has a convergent subsequence and by our hypothesis (and Bolzona Weierstrass Theorem). Also, theorem 2.5.2 states, "subsequences of a convergent sequence converge to the same limit as the original sequence". So, (I_{n}) also must converge to zero.

    From here I am not sure what to do. We just got into Cauchy sequences and how to do basic proofs with some Cauchy theorems. I am not sure if I need to use any of those theorems to prove the above.

    I would appriciate some guidence. I am having a really hard time with this one.
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  2. #2
    MHF Contributor
    Opalg's Avatar
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    I would look at the sequences of endpoints. The left-hand endpoints a_n form an increasing sequence of numbers that is bounded above. The right-hand endpoints b_n form an decreasing sequence of numbers that is bounded below. Show that these two sequences converge to the same limit c. Then deduce that \textstyle\bigcap I_n = \{c\}.
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