Suppose $\{X_i:i\geq 1\}$ are i.i.d. with mean $\mu$. By the strong law of large numbers, $\bar{X}_n \stackrel{a.s.}{\rightarrow}\mu$. Does this imply the following?
There exists a $\delta>0$, such that for large enough $n$, $$ P(\bar{X}_n\in [\mu-\delta, \mu+\delta]) = 1. $$
In general does this statement hold true if any estimator $\widehat{\theta}_n$ converges almost surely to a parameter $\theta_0$? Any ideas will be appreciated.