divergence of sum ∑a_i+∑1/b_i=∞ based on two sequences ⇒ divergencence of ∑x_i/(x_i^2+1)=∞ based on combined sequence [closed]

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$$\lim a_i = \infty\text{,}$$$$\lim b_i = 0.$$If then $\sum_{i=1}^{\infty} a_i + \sum_{i=1}^{\infty}1/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. Any ideas welcome, kind regards!