A few notations about Taylor Expansion in Empirical Likelihood by Owen.

Question

I read the proof of Wilk's Theorem in Art B. Owen's Book: Empirical Likelihood, but found a place I did not understand. I think it is quite a mathematical analysis problem. On page 221, $\log(1+Y_{i})=Y_{i}-\frac{1}{2}Y_{i}^{2}+\eta_{i}$, where for some fintie $B>0$, $$P(\vert\eta_{i}\vert\leqslant B\vert Y_{i}\vert^{3}, 1\leqslant i\leqslant n)\rightarrow 1,$$ as $n\rightarrow\infty$. I know the log thing is expanded by Taylor Series, but how can I understand the convergence in probability in the end?