It is known that $c_{0}$ is a Banach space with norm $\lVert x \rVert_{\infty} = \sup_{k \in \mathbb{N}} \lvert x_{k} \rvert$.
When proving that dual space of $c_{0}$ is $l^{1}$ is common to show that every functional $f$ in $c_{0}^{*}$ has the form $$ f(x) = \sum_{k = 0}^{\infty} x_{k}f(e_{k}), \quad (f(e_{k}))_{ k \in \mathbb{N}} \in l^{1}, $$
where $e_{k}$ is the sequece that has 1 in position $k$ and 0 elsewhere. But to do this, it is essential to know that every $x = (x_{k})_{k\in \mathbb{N}}$ in $c_{0}$ can be represented as the convergent series (in the norm $\lVert \cdot \rVert_{\infty}$) $\sum_{k = 1}^{\infty} x_{k} e_{k}$ in order to use the linearity and continuity of $f$ . But this fact is not so trivial for me. Thanks for suggestions.
