I tried first using the fact that $c_0$ is Banach to apply the Uniform Boundedness Principle on the function series $(T_n)_n = \{\sum_{k=1}^n a_kx_k\}$, $T_n:c_0 \rightarrow \mathbb{K}$, and then to extend that results to a series of functionals $T_n:l_\infty \rightarrow \mathbb{K}$ using Hahn-Banach theorem, but that way I can't get absolute convergence of (a_n)_n, only "normal" convergence, which is not enough for $(a_n)_n \in l_1$.
Perhaps there's a better way?