Lim Sup question

**hmmmm** · October 22nd 2011, 12:36 PM

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

**mohammadfawaz** · October 23rd 2011, 09:35 PM

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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