Translation of this exercise from Dutch

Question

I would like to get this text translated from Dutch to English:

I tried using Google translator but the result is confusing me:

Answer 1

• Given a bounded sequence $(y_n)_n$ in $\mathbb{C}$. Show that for every sequence $(x_n)_n$ in $\mathbb{C}$ for which the series $\sum_n x_n$ converges absolutely, that also the series $\sum_n \left(x_ny_n\right)$ converges absolutely.

• Suppose $(y_n)_n$ is a sequence in $\mathbb{C}$ with the following property: for each sequence $(x_n)_n$ in $\mathbb{C}$ for which the series $\sum_n x_n$ converges absolutely, also the series $\sum_n \left(x_ny_n\right)$ converges absolutely. Can you then conclude that $(y_n)_n$ is a bounded sequence? Explain!

Answer 2

(a) Let $(y_n)_n$ be a bounded sequence in $\mathbb C$. Show that for every sequence $(x_n)_n$ in $\mathbb C$ for which the series $\sum_nx_n$ converges absolutely also the series $\sum_n(x_ny_n)$ will converge absolutely.

(b) Now assume that $(y_n)_n$ is a sequence in $\mathbb C$ such that for every sequence $(x_n)_n$ for which series $\sum_nx_n$ converges absolutely also the series $\sum_n(x_ny_n)$ converges absolutely. Does this imply that $(y_n)_n$ is bounded? Give an argumentation of your answer.

So actually in (b) you are asked whether the converse of (a) is true or is not true.