Almost complete convergence

Question

In the book nonparametric functional data analysis, page 232, i don't understand why: The almost complete convergence of $Y_n$ to $l\ne 0$ implies that there exists some $\delta > 0$ (choose for instance $\delta = l/2$) such that

$$\sum_{i\ge 0}\mathsf{P}(|Y_n| \le \delta) < \infty.$$ Cordially.

Answer 1

Suppose that $l>0$. Since \begin{align} \mathsf{P}(|Y_n|\le l/2)&=\mathsf{P}(-l/2\le Y_n\le l/2)\\ &\le \mathsf{P}(Y_n\le l-l/2)\le \mathsf{P}(|Y_n-l|\ge l/2), \end{align}

$$ \sum_{n\ge 1}\mathsf{P}(|Y_n|\le l/2)\le \sum_{n\ge 1}\mathsf{P}(|Y_n-l|\ge l/2)<\infty. $$