
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

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 subsequence 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.