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Thread: Lim Sup question

  1. #1
    Senior Member
    Dec 2008

    Lim Sup question

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

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

    lim_{n\rightarrow \infty}\ sup_{x\in X} |f_n(x)|=0 as 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
    Feb 2010
    Lebanon - Beirut

    Re: Lim Sup question

    The question is not very clear. However, there is one problem in the third line: lim_{n\rightarrow\infty}y is exactly y, because you're chosing y as one function from the sequence 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 limsup is not zero. If that's what you mean by y then your statement is correct.
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