Let $(X_n)_{n\ge 1}$ be a sequence of random variables such that $X_n \longrightarrow x_0$ almost surely, where $x_0\in [\alpha, 1-\alpha]$ and $0<\alpha<1$. I'm trying to show that for all $\epsilon >0$, $$ P\left(|X_n - x_0|> \frac{\alpha}{2}\right) < \epsilon\quad\quad (*) $$ as $n\to +\infty$.
My attempt: since $X\longrightarrow x_0$ a.s, we have for all $\epsilon'>0$ and $\epsilon >0$, $\forall n\ge n_{\epsilon'}$ $$ P\left(|X_n - x_0| > \epsilon' \right) < \epsilon $$ Then we choose $\epsilon' = \alpha/2$ and we will get $(*)$. The solution is quite simple but I'm not sure! What do you think about this? Thank you for any comment or suggestion.