Consider ($\tau_n$) a diverging sequence of stopping times (e.g. $\inf\{t: X_t>n\}$). We can write the stopped local martingale $X_t^{\tau_n}$ = $X_{t\wedge \tau_n}$, which yields $\lim_{n\rightarrow +\infty} X_t^{\tau_n} = X_t$
I was solving some exercises and saw some people using $E[X_t] = E[\underline{\lim}X_t^{\tau_n}]$, where $\underline{\lim}$ is the liminf.
This equality is not clear for me (the use of $\inf$), and any help on the intuition behind it would be greatly appreciated.