Linear functionals on space of all converges sequences.

Suppose that $V$ is the space of all convergent real sequences, $(x_i)_{i=0}^{\infty}$

How to show that every absolute convergent series, $\sum_{i=0}^{\infty}a_i$ defining a linear functional with:

$$\phi(x_i)=\sum_{i=0}^{\infty}a_ix_i$$

First of all:

$$\left|\sum_{k=0}^\infty a_kx_k\right|\le M\sum_{k=0}^\infty a_k$$

with $\,M\,$ such that $\,|x_k|\le M\,\,\,,\;\;\forall\,k\in\Bbb N\,$ (why such M exists?) , so the function $\,\phi(\{x_k\})\,$ is well defined as the series defining it converges absolutely. Now you only have to check that

$$\phi(\{x_k\}+\{y_k\})=\phi(\{x_k\})+\phi(\{y_k\})$$

$$\phi\{tx_k\})=t\,\phi(\{x_k\})\;,\;\;t\in\Bbb R$$

Pay attention to the fact that unless the series $\,\sum a_k\,$ is the zero series, the lin. functional $\,\phi\,$ is non zero (and thus it is onto...), and then its kernel is a hyperplane (=maximal proper linear subspace) of the space of all finitely convergent sequences.