If $\sum c_n$ is convergent where $c_n = a_1b_n + \cdots +a_nb_1$, then $\sum c_n = \sum a_n\sum b_n$

$ \sum a_n$ and $ \sum b_n$ is convergent and $ c_n = a_1b_n + \cdots + a_nb_1$ . Prove that if $ \sum c_n$ is convergent then $ \sum c_n = \sum a_n \sum b_n$ .

This question has been answered using Abel’s Lemma in this post. But I am looking forward to a more basic solution which only uses basic definition and Cauchy’s convergence test.