By hypothesis, the "squared" infinite series $\sum_{n=0}^\infty a_n^2$ and $\sum_{n=0}^\infty b_n^2$ are both convergent, say to the limits $\alpha$ and $\beta.$
By the Cauchy-Schwarz inequality and the triangle inequality we have for the partial sums the relation $$\left|\sum_{n=0}^N a_n b_n\right|\le \sum_{n=0}^N |a_n b_n|\le \left(\sum_{n=0}^N a_n^2 \right)^{\frac 12}\left(\sum_{n=0}^N b_n^2 \right)^{\frac 12}$$ for all $N\ge 0.$
