How prove exists a sequence $\{a_{n}\}$ of real numbers such that $\sum_{n=1}^{\infty}a^2_{n}<\infty,\sum_{n=1}^{\infty}|a_{n}b_{n}|=\infty$ Suppose that the series $\displaystyle\sum_{n=1}^{\infty}b^2_{n}$ of postive numbers diverges. Prove that
 there exists a sequence $\{a_{n}\}$ of real numbers such that
$$
\sum_{n=1}^{\infty}a^2_{n}<\infty
\quad\text{and}\quad
\sum_{n=1}^{\infty}|a_{n}b_{n}|=\infty.
$$
My try: maybe this Cauchy-Schwarz inequality have usefull
$$\Big(\sum_{n=1}^{\infty}a^2_{n}\Big)\Big(\sum_{n=1}^{\infty}b^2_{n}\Big)\ge
\Big(\sum_{n=1}^{\infty}a_{n}b_{n}\Big)^2$$

Let $S_N= \sum_{n=1}^N b_n^2$ and take $a_n=b_n/\sqrt{S_N}$.  Then 
$$\sum_{n=1}^N a_n^2 = 1$$
and
$$\sum_{n=1}^N a_n b_n = \sqrt{S_N}$$.
Let $N\to\infty$.
