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Thread: Lim Sup question

  1. #1
    Senior Member
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    Dec 2008
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    Lim Sup question

    If I want to know wither $\displaystyle lim_{n\rightarrow \infty}\ sup_{x\in X} |f_n(x)|$=0 can i say that if for some value of
    $\displaystyle y=|f_n(x)|\ \mbox{where}\ x\in X$ we have that

    $\displaystyle lim_{n\rightarrow \infty} y>0$ then it is not the case that

    $\displaystyle lim_{n\rightarrow \infty}\ sup_{x\in X} |f_n(x)|$=0 as $\displaystyle y\leq sup_{x\in X} |f_n(x)| $

    Im struggling to come up with a counter example or anything for this.

    thanks for any help
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  2. #2
    Member mohammadfawaz's Avatar
    Joined
    Feb 2010
    From
    Lebanon - Beirut
    Posts
    100

    Re: Lim Sup question

    The question is not very clear. However, there is one problem in the third line: $\displaystyle lim_{n\rightarrow\infty}y$ is exactly $\displaystyle y$, because you're chosing $\displaystyle y$ as one function from the sequence $\displaystyle f_n$. This doesn't make sense. On the other hand, If you can find a sub-sequence of functions with limit greater than zero, then you know that your $\displaystyle limsup$ is not zero. If that's what you mean by $\displaystyle y$ then your statement is correct.
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