If $x_k ≥ 0\;\forall \in \mathbb N$, and $y_k$ a bounded sequence, then the series $\sum_{k=1}^\infty x_ky_k$ converges

Question

Hi I'm really struggling with this proof.

For a start I'm struggling to believe it's true:

For example, if we take $x_k = \dfrac{1}{k^2}$ and $y_k = -k^3$ (which is bounded above by any positive number), then the series $x_ky_k$ does not converge?

What am I doing wrong? I feel like I'm being insanely stupid.

Thank you!

Answer 1

The result is true for a bounded sequence $(y_k)$ i.e. $$\exists M>0,\quad \forall k\;\; |y_k|\le M$$ and fails if $(y_k)$ is just bounded below or bounded above.