Convergence of series/absolute convergence Let $y_n$ be a sequence of real numbers such that for all sequences of real numbers $x_n$ with $\lim x_n =0$  the series $\displaystyle \sum_{n=1}^{\infty} x_n y_n $ converges. Prove that $\displaystyle \sum_{n=1}^{\infty} |y_n| $ converges.

I deleted my solution... All i know/extract from the exercise is that $\lim x_n y_n =0$ and $\lim x_n =0$. Now I'm stuck. I don't know how to continue. Any hints appreciated. I know many criterions/ tests that should help me examine if the series converges, yet I cannot apply them.. because I do  not know if $\lim |y_n| =0$. If it does not the series clearly diverges.
 A: Here is how to see $\lim |y_n| =0$: if the sequence of non-negative terms $|y_n|$ does not converge to $0$, then there is $\epsilon > 0$ and a sequence of indices $n_1 < n_2 < \ldots$, such that $|y_{n_k}| > \epsilon$ for $k = 1, 2, \ldots$. Define a sequence $x_n$ such that
$$x_{n_k} = \frac{\mathrm{sgn}(y_{n_k})}{k} $$
and $x_n$ is $0$ if $n$ is not one of the $n_k$. Then $\lim x_n = 0$, but $\sum_{n=1}^{\infty}x_ny_n$ is not convergent, as the subsequence comprising its non-zero elements is bounded below by the harmonic sequence $\frac{\epsilon}{k}$. This contradicts our assumptions, so we must have $\lim|y_n| = 0$.
To answer the main question, assume $\sum_{n=1}^{\infty}|y_n|$ does not converge, then there is a sequence of indices $N_1< N_2 < \ldots$ such that the partial sum $\sum_{n=1}^{N_k}|y_n| > k$, $k = 1, 2 \ldots$. Now define
$$
x_n = \frac{\mathrm{sgn}(y_{n})}{\sqrt{k_n}}
$$
where $k_n$ is the least $k$ such that $n \le N_k$. then $\lim x_n = 0$, but $\sum_{n=1}^{N_k}x_ny_n > \frac{k}{\sqrt{k}} = \sqrt{k}$, so $\sum_{n=1}^{\infty}x_ny_n$ is not convergent, contradicting our assumptions.
