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Thread: Simple sequences problem

  1. #1
    Nov 2008

    Simple sequences problem

    I came across something used in another proof and I wanted to prove this simple fact for myself, although when I tried to pin is down with epsilons and the like, I found that I kept getting stuck! Considering sequences in \mathbb{R}. I would like to show that if \frac{x_{n+1}}{x_n}\rightarrow 0 then x_n \rightarrow 0.

    I would only like a hint with regards to how to do it, as I want to do it myself!

    If I could prove a slightly weaker conclusion, that \{x_n\} converges, then I can use the fact that convergent implies bounded and then I'd have something like

    |x_{n+1}| < \epsilon |x_n|

    for suitably large n, then I could use the bound on x_n to force x_{n+1} to zero and therefor \{x_n\} \rightarrow 0.

    Am I over thinking this? Is there a simple proof? Some clever trick I have not seen?

    Any help would be much appreciated, thanks
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  2. #2
    MHF Contributor

    Aug 2006
    Is clear to you that from the given we have \left| {\dfrac{{x_{n + 1} }}{{x_n }}} \right| \to 0?
    Therefore, we know that the series \sum\limits_n {x_n } converges.
    That implies  \left( {x_n } \right) \to 0.
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