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Math Help - Constant vs Monotonic Sequences

  1. #1
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    Constant vs Monotonic Sequences

    I'm having a few problems with an assignment about sequences. I'm supposed to decide if these statements are true or false (if true, give a short proof; if false, give a counter-example). Here are the statements:

    a) Every constant sequence is monotonic.
    b) Every monotonic sequence is constant.
    c) Every increasing sequence is unbounded.
    d) Every strictly increasing sequence is unbounded.

    After reading through all of them, they all appear to be false. For (a) and (b), the definition of a monotonic sequence is one that is increasing OR decreasing, and the definition of a constant sequence is one the is increasing AND decreasing. I think this definitely makes (b) false, but I'm questioning if that's the same for (a) as well. Am I comparing the definitions the wrong way?

    For (c) and (d), take for example a(n) = -1/n. This sequence gets larger for all n within natural numbers, but it is also has clear bounds. Its lower bound would be -1 when n=1, and it's upper bound would be 0. So this is a strictly increasing sequence (which would also cover a regular increasing sequence) that is bounded. This would prove that (c) and (d) are false. Does this make sense?
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  2. #2
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    The answer to these strictly depends upon the exact wording of your definitions. We don’t have that information.
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  3. #3
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    Here's the definitions exactly way they were given (note: read a(n) as "a sub n", as I'm not aware of how to write subscripts here):

    A sequence (a(n)) is increasing if a(n+1) >= a(n) for all n within natural numbers.
    A sequence (a(n)) is strictly increasing if a(n+1) > a(n) for all n within natural numbers.
    We say (a(n)) is monotonic if it is either increasing or decreasing.
    A sequence (a(n)) which is both increasing and decreasing is a constant sequence (i.e. there exists an a within real numbers so that a(n) = a for all n within natural numbers).
    A sequence (a(n)) is bounded above if there exists a U within real numbers such that a(n) <= U for all n within natural numbers.
    A sequence (a(n)) is bounded below if there exists a L within real numbers such that a(n) >= L for all n within natural numbers.
    If (a(n)) is bounded below and above, we say it is bounded.
    If (a(n)) is not bounded, it is unbounded.
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  4. #4
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    Using these definitions statement (a) is true.
    If P \text{ and } Q is true then P \text{ or } Q is also true.
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  5. #5
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    I agree with your reasoning about (c) and (d). Check, however, whether 0 is a natural number.
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