Let $\ell^\infty$ be the Banach space of real bounded sequences with its usual norm and $S\subset\ell^\infty$ be the space of convergent sequences. Define $f:S\rightarrow\mathbb{R}$ by $f(x)=\lim_{n\rightarrow\infty}x_n$
where $x=(x_1,x_2,...,x_n,...)$. Extend $f$ to a bounded linear functional $F:\ell^\infty\rightarrow\mathbb{R}$ by using Hahn-Banach Theorem. Does $F$ belongs to $\ell^1$?