Suppose we have a family (indexed by $n$) of discrete-time random process $\{X_{t}^n\}, {t\geq0} $, taking values in $\mathbb{R}$.
For each $n,t$, $X_{t}^n\geq0$. And $X_{t}^n$ converges pointwise towards a discrete-time random process $X_{t}$.
Define the stopping times correspondingly, $A$ is a constant: $T_n = \inf[t:X_{t}^n\geq A]$, $T = \inf[t:X_{t}\geq A]$. (Sorry I cant type "{" for some unknown reason)
My ultimate goal is to give a lower bound for $\liminf\limits_{n\to\infty} E\{T_n\}$, and now I have already had a lower bound for $E\{T\}$. I am wondering how to relate $E\{T\}$ and $\liminf\limits_{n\to\infty} E\{T_n\}$?
It is not true in general that $T_n$ converges to $T$. Consider for example the deterministic case $x = (1,0,0,\ldots)$ and $x^n = (1-1/n, 0, 0,\ldots)$. Then taking $A = 1$ we have $T = 0$ but $T_n = +\infty$.
However, it is true that $T \leq \liminf_n T_n$. To prove this, notice that by definition of $T$, we have $X_t < A$ for every $0\leq t < T$. Since $(X_0^n, X_1^n, \ldots, X_{T-1}^n) \to (X_0,X_1,\ldots, X_{T-1})$ almost surely, there exists $N= N(\omega)$ such that for every $n \geq N$, $X_t^n < A$ for every $0\leq t < T$. By definition of $T_n$, it follows that $T_n \geq T$ for every $n \geq N$. In particular, $T\leq \liminf_n T_n$.
Now you can apply Fatou's lemma.