Let $X_n \geq 0$ be an iid sequence with all moments finite. Let $E_n \geq 0$ is a increasing sequence of random variables with $\lim_{n \rightarrow \infty}E_n = \infty$. If $Y_n = min\{X_n,E_n\}$. then is $Y_n \neq X_n$ only finitely many times.