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 then convergent $ sum c_n = sum a_n sum b_n $.

This question was answered using Abel Lemma in this post. But I'm looking forward to finding a simpler solution that uses only Cauchy's basic definition and convergence test.