Decomposition of a linear functional. 
Let $C = \left\{ \{x_i\} \in l_\infty | \exists \lim_{i \rightarrow \infty} x_i \right\}$. Let $\phi \in C^{*}$ ($C^{*}$ is a dual space). Let $y=\{y_i\} \in l_{\infty}$ $\phi_y: \{x_i\} \rightarrow \sum_i x_i y_i$. Prove that $\exists \alpha \in \mathbb{R}, \exists y \in l_{\infty}$ such that $\phi = \phi_y + \alpha * \lim$

I understand all of the definitions and conditions of the problem, but i don't know enough tools that can help me for solving this.
 A: For $k \in  \mathbb{N}$, let ${e}^{k} = {\left\{{{\delta}}_{i}^{k}\right\}}_{i \in  \mathbb{N}}$ where ${\delta}$ is the Kronecker delta. Let
${y}_{k} = {\phi} \left({e}^{k}\right)$. Let $M = {\left\|{\phi}\right\|}_{{C}^{\ast }}$ and let ${u}^{n} = \sum _{k = 0}^{n} \text{sgn} \left({y}_{k}\right) {e}^{k} \in  C$.
We have
$$\sum _{k = 0}^{n} \left|{y}_{k}\right| = \sum _{k = 0}^{n} \text{sgn} \left({y}_{k}\right) {\phi} \left({e}^{k}\right) = {\phi} \left({u}^{n}\right)  \leqslant  M {\left\|{u}^{n}\right\|}_{C}  \leqslant  M$$
Hence $y = {\left\{{y}_{i}\right\}}_{i \in  \mathbb{N}} \in  {\ell }_{1}$ and ${\left\|y\right\|}_{{\ell }_{1}}  \leqslant  {\left\|{\phi}\right\|}_{{C}^{\ast }}$. It follows that
the expression
$${{\phi}}_{y} \left(x\right) = \sum _{i \in  \mathbb{N}} {y}_{i} {x}_{i}$$
defines a bounded linear functional on $C$.
We claim that if $x \in  C$ and $\lim  x = 0$, then
${\phi} \left(x\right) = {{\phi}}_{y} \left(x\right)$. Indeed the condition imply that
${v}^{n} \rightarrow  x$ in $C$ where ${v}^{n} = \sum _{k = 0}^{n} {x}_{k} {e}^{k}$.
But one has ${\phi} \left({v}^{n}\right) = {{\phi}}_{y} \left({v}^{n}\right)$ and as ${\phi}$
and ${{\phi}}_{y}$ are continuous, the result follows.
Now let $x \in  C$ be such that $\lim  x = {\lambda}$
one can write $x = \left(x-{\lambda} w\right)+{\lambda} w$ where
$w = {\left\{1\right\}}_{i \in  \mathbb{N}}$ is the constant sequence equal to $1$. We have
$${\phi} \left(x\right) = {\phi} \left(x-{\lambda} w\right)+{\lambda} {\phi} \left(w\right) = {{\phi}}_{y} \left(x-{\lambda} w\right)+{\lambda} {\phi} \left(w\right) = {{\phi}}_{y} \left(x\right)+\left({\phi} \left(w\right)-\sum _{i \in  \mathbb{N}} {y}_{i}\right) \lim  x$$
