I'm trying to solve the following question.
Prove that there exists a bounded linear functional $F:l^{\infty}\rightarrow\mathbb R$ satisfying the following conditions:
(i) $|F(x)|\leq \sup_{n} |x_n|$
(ii) $F(x)=\lim_{n}x_n$, if limit exists
(iii) $\liminf x_n\leq F(x)\leq \limsup x_n$.
I proved the first two parts by defining the functional $F$ for the space of convergent sequences and then extending it to the whole of the Banach space $l^{\infty}$ using the Hahn-Banach theorem. However, I can't prove part (iii). Does it follow simply by using the definitions of $\liminf$ and $\limsup$? This seems to be not too difficult, but I can't seem to get this part. I'm not sure this particular part needs any functional analysis and so the tag need not be justified. Thank you for any help.