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Math Help - Sequence-Constant Inequalities

  1. #1
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    Sequence-Constant Inequalities

    If you have a sequence x_{n} and an inequality
    (|x_{n}|)^{\frac{1}{2}}<\beta

    Does that mean that the point to which the sequence converges (if it does) is less than \beta or does that mean (|x_{n}|)^{\frac{1}{2}}<\beta, \forall n?
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  2. #2
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    Quote Originally Posted by miatchguy View Post
    If you have a sequence x_{n} and an inequality
    (|x_{n}|)^{\frac{1}{2}}<\beta

    Does that mean that the point to which the sequence converges (if it does) is less than \beta or does that mean (|x_{n}|)^{\frac{1}{2}}<\beta, \forall n?

    In a rather unsurprising fashion, (|x_{n}|)^{\frac{1}{2}}<\beta means just (|x_{n}|)^{\frac{1}{2}}<\beta . As simply as that.

    They could tell you this inequality is true for any n or only for some (one, a few, infinite, all...) n's, but that's something you can't ask here but rather ask the person who gave you the inequality.

    Now, if x_n\xrightarrow [n\to\infty]{}\alpha then |x_n|\xrightarrow [n\to\infty]{}|\alpha| , but nothing else can be said about the relation between \alpha,\,\beta until we know more about the given inequality.

    Tonio
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