# Existence of a $l_ {\infty} ^*$ element [duplicate]

Exercise: Prove there exist a bounded linear functional $L :l_{ \infty} \rightarrow \mathbb{R}$ such that for every $(x_n)=x \in l _ { \infty }$ $$\lim \mathrm{Inf} (x_n) \leq L(x) \leq \lim \mathrm{Sup} (x_n).$$ My current progress is that should be $L \in l _{ \infty}^*\setminus l_1$. Also I know since $l_1$ is not bidual so there are such functionals. Any help is appreciated.

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@Norbert I may missing something obvious, but why such functional implies the above inequalities? –  clark Jul 4 '12 at 18:41
here is a proof homepages.math.uic.edu/~furman/math569/Banach-LIM.pdf –  Norbert Jul 4 '12 at 18:44
thanks that's seems intersting! –  clark Jul 4 '12 at 18:50
Not at all ;)${}$ –  Norbert Jul 4 '12 at 18:50