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Math Help - Question

  1. #1
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    Question

    There is a question that I have been having hard time finding a formal solution. It's an easy question I don't know why I have been having hard time solving it, but here it is.

    (an) is a non decreasing sequence.
    Prove that if the sequence is not bounded then it converges to infinity.

    So first I tried to assume that it doesn't converge to infinity and somehow get a contradiction to the fact that it's a non decreasing sequence or that it is bounded but I failed.

    So I said that , if the sequence is not bounded that means for every M in R (Real numbers) there exists an N such that M<aN
    It's a non decreasing sequence therefore for every n in Natural numbers that n>N we have M<aN<an So we get that for every M in R there exists an N such that for every n>N we get M<an and therefore it satisfies the definition of the limit (for infinity) Therefore it converges to infinity.

    So can we count what I wrote as a proof?

    My second question is, if it is correct, what other ways could I have used to prove? I got a little bit confused with the contrapositive here. Could I have assumed that it doesn't converge to infinity and somehow get a contradiction?

    Thank you.
    Last edited by davidciprut; April 3rd 2014 at 03:31 AM.
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  2. #2
    MHF Contributor
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    Re: Question

    diverges to infinity. sequences don't converge to infinity.

    Other than that wording your proof looks correct.

    For your second question the answer is yes. If you assumed that $a_n$ converged to some value you could show that contradicts $a_n$ being unbounded but there is really no need for that here. Your proof is essentially 2 lines.
    Thanks from davidciprut and LimpSpider
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