Let $\{a_n\}_{n \in \mathbb{N}} \in \mathbb{C}$ and $\{b_n\}_{n \in \mathbb{N}} \in \mathbb{C}$ and $\{c_n\}_{n \in \mathbb{N}} \in \mathbb{C}$ be three complex sequences and let $x_n = |a_n + b_n|$ such that $$ \lim_{n \to \infty}a_n = \lim_{n \to \infty}b_n = \lim_{n \to \infty}c_n = \lim_{n \to \infty} \dfrac{b_n}{a_n} = \lim_{n \to \infty} \sum_{p=0}^n x_p x_{n-p} = 0 $$ I would like to know if is true that $$ \lim_{n \to \infty} \dfrac { \sum_{p=0}^n x_p x_{n-p} } { |c_n| } = \lim_{n \to \infty} \dfrac { \sum_{p=0}^n a_p a_{n-p} } { |c_n| } $$
Thanks for any suggestion