Bounding the series $\sum \frac{u_i u_j}{j-1}$

I was playing with some series, and came up with the following problem (which I am not sure is true):

Let $(u_i)$ be a sequence of positive reals. Furthermore, suppose that $\sum u_i ^2 $ converges. Is it true that $\sum_{1\le i <j} \frac{u_i u_j}{j-1}$ converges? Furthermore is it true that: $\sum_{1\le i <j} \frac{u_i u_j}{j-1} \le (2-\frac{\pi^2}{12})\sum u_i^2 ?$

Using Cauchy-Schwarz inequality, I found that it is indeed true, that the series converges, but I couldn't prove the inequality, and the bound I found was too large