Prove that if $ \sum{a_n^2}$ and $\sum{b_n^2}$ is convergent infinite series then $\sum{a_n.b_n}$ is absolutely convergent

Question

if we are using Cauchy general theorem of convergence then we have $$|a_{n+1}^2+a_{n+2}^2+.....a_{n+p}^2| < \varepsilon\;\text{ for all } n\ge m_1\;\text{ and }\;p\ge1 $$

$$|b_{n+1}^2+b_{n+2}^2+.....b_{n+p}^2| < \varepsilon\;\text{ for all } \;n\ge m_2\;\text{ and }\;p \ge 1. $$

Now to prove, consider $$|a_{n+1}b_{n+1}|+|a_{n+2}b_{n+2}|+\dots+|a_{n+p}b_{n+p}|.$$

After this I unable to guess the substitution that to make, please help. How should I proceed further?