divergence of split sequences with $\sum a_i + \sum b_i = \infty \Rightarrow \sum x_i/(x_i^2 + 1) = \infty$

A given series $(x_i)_{i=1}^{\infty}$, is split into $(a_i)$ and $(b_i)$ with $(x_i) = (a_i) \cup (b_i) ,$ such that $\lim a_i = \infty,$ and $\lim b_i = 0.$

If then $\sum_{i=1}^{\infty} a_i + \sum_{i=1}^{\infty}b_i = \infty $, then follows $\rightarrow \sum_{i=1}^{\infty} \frac{x_i}{x_i^2 + 1} = \infty.$

I got this from a research paper. So far, I proved the other direction, which was easy, but this problem confuses me a bit, I didn't meet this splitting-type of problems.