Let $\{X_n\}$ be some sequence of positive random variables. Then what is $$ \lim_{n\rightarrow \infty}\mathbb{E}\left[\sup_{\theta} \mathbb{1}\left\{ \frac{n-1}{n} \theta \le X_n < \theta \right\} \right]? $$
What I have done.
Since for each realization of $X_n$, we can set $\theta$ by $\frac{n}{n-1}X_n$, then we have $$ \sup_{\theta} \mathbb{1}\left\{ \frac{n-1}{n} \theta \le X_n < \theta \right\} = 1\qquad\text{a.s.} $$ Thus, the limit value is $1$. So am I right?