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Since $a_n$ converges then its sequence of partial sums $(x_n)$ converges to a limit x. Also, since $b_n$ converges then its sequence of partial sums $(y_n)$ converges to a limit y. Observe that by using the multiplication limit theorem: If $x_n \to x$ and $y_n \to y$ then $x_n*y_n \to x*y$. We know that the product of its partials sums converge, thus the product of the series converges.$
Is this correct? Any guidance is appreciated !