Continuous Linear Functional on $\ell^{\infty}$ I'd like help answering two questions.
1) Prove that there is a continuous linear functional on $\ell^\infty$ such that $f(e_n)=0 \ \forall n \in \Bbb{N}$ and $f(a)=5$ where $a=(1,1,1,1,1,1,\ldots)$.
2) Prove that there is not a continuous linear functional on $\ell^\infty$ such that $f(e_n)=0 \ \forall n \in \Bbb{N}$ and $f(a)=4$ for $a=(1,1/2,1/3,1/4,\ldots)$.
Note: $e_n$ stand for the point $(0,0,\ldots,0,1,0,\ldots,0,0)$ with $1$ in the $n$-th position and the rest $0$'s, i.e $(e_n)=(\delta_{mn})$.
Thanks in advance.
 A: The key observation (for both questions) is that the Banach subspace of $\ell^\infty$ generated by $\{e_1,e_2,\ldots\}$ is $c_0$. 
A: There are positive continuous functionals on $\ell_\infty$, which map every convergent sequence to its limit. Banach limits are one class of such functionals, see fore example this question. Another example is a limit of a bounded sequence along an ultrafilter, see Wikipedia, this answer or this question.
For the second part just notice that if $f(e_n)=0$, then also each $x_n=(1,1/2,1/3,\dots,1/n,0,0,\dots,0,\dots)$ is mapped to zero. Since $f(x_n)=0$ and $a_n\to a$ in $\ell_\infty$, we get $f(a)=0$.
A: For question 1), you could provide an explicit example of such a functional.  One such example would be $f:\ell^\infty\to \mathbb{R}$ given by
$$
f(\{x_n\}) = 5 \cdot \lim_{n\to\infty}\frac 1n \sum_{k=1}^n x_k
$$
For the purposes of your question, it suffices to prove that $f$ is a bounded (i.e. continuous) linear functional satisfying the necessary criteria where it is defined.  From there, one must use the HB theorem to extend $f$ to give a functional over $\ell^\infty$.
