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Monotonic Sequence Theorem Question

  1. Sep 18, 2004 #1
    Suppose you know that [tex]\left{ a_n \right}[/tex] is a decreasing sequence and all its
    terms lie between the numbers 5 and 8. Explain why the sequence has a limit. What can
    you say about the value of the limit?

    My Answer:

    This sequence has a limit because it is both bounded and monotonic, as it is stated in the
    Monotonic Sequence Theorem. The minimum value of this particular limit must be 5.

    My question:

    Is that thoroughly answered? Did I miss anything?
  2. jcsd
  3. Sep 18, 2004 #2
    Well, you can say that the limit is greater than or equal to five. As you know, a certain theorem says that any bounded, monotonic sequence has a limit. But if the problem is stated as you have written it, I don't think you can say that the limit must be 5.

    All of the terms lie between 5 and 8. But do you know if the infimum of the sequence is 5? If the infimum is 5, then yes, the limit is also 5. But if 5 is simply a lower bound (and not necessarily the infimum), then you can't say that the limit is 5.
  4. Sep 18, 2004 #3
    I think get it. The limit is not necessarily 5.

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