# 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.