Let $(x_n)$ and $(y_n)$ be sequences such that the sequence of partial sums of the series $\sum_{n=1}^\infty{x_n}$ is bounded, and furthermore the series
$\sum_{n=1}^\infty{|y_n - y_{n+1}|} $ converges, while $y_n \to 0 $ for $x \to \infty$
Prove that the series $\sum_{n=1}^\infty{x_n{y_n}}$ converges.
Hint: use the summation by parts formula