Theorem. If $\sum_{n=1}^{\infty}a_n$ converges absolutely and $(b_n)$ is a bounded sequence, then $\sum_{n=1}^{\infty}a_nb_n$ converges absolutely.
Proof. Suppose that $\sum_{n=1}^{\infty}a_n$ converges absolutely and $(b_n)$ is a bounded sequence. The second hypothesis means that there exists at least one positive real number $M$ such that \begin{align*} &0\leq |b_n| < M,\hspace{0.3 in} \forall n\in\mathbb{N}\\ \tag{1} &\implies 0\leq |a_nb_n|\leq M|a_n|,\hspace{0.3 in} \forall n\in\mathbb{N}. \end{align*} The first hypothesis implies that the series $\sum_{n=1}^{\infty}M|a_n|$ converges, which, togheter with the claim at $(1)$, implies, by the Comparision criteria, that the series $\sum_{n=1}^{\infty}|a_nb_n|$ converges, which means that the series $\sum_{n=1}^{\infty}a_nb_n$ converges absolutely. $\blacksquare$
Is my proof correct? Thank you so much for your help.