Let $c=(c_n)subset mathbb{C}$ and suposse that $sum_{n=1}^{infty} a_n c_n$ converges for all $a=(a_n)in l^1(mathbb{N})$. Show that $c$ is bounded.

Hi. I cannot conclude on this problem. I attach the possible ideas I have.

Idea: For all $k$, $|c_k|leq sum_{n=1}^{k} |a_n||c_n|leq sum_{n=1}^{infty} b_nc_n$ with $a=(frac{overline{c_1}}{|c_1|},ldots, frac{overline{c_k}}{|c_k|},0,ldots, )in l^1(mathbb{N})$

and I would like to consider $b$ as $b=(frac{overline{c_1}}{|c_1|},ldots, frac{overline{c_k}}{|c_k|},ldots)$ but this $bnotin l^1(mathbb{N})$… and that does not allow me to obtain a bound for $| c_k|$