Let, $X_n$ converges $X$ in distribution. Give a counterexample of this statement: $\limsup P[X_n\leq x]\leq P[X\leq x]$
I know $\limsup P[X_n\leq x]\leq P[\limsup X\leq x]$. Then if I can give an example so that $X_n$ converges $X$ in distribution but $\limsup X_n\neq X$, it is done. But could not find one. Thanks for any help.
There is no counter example to the above statement because of the Portmanteau theorem. Specifically, the Portmanteau theorem says that weak convergence of probability measures $\mu_n$ to $\mu$ is equivalent to the statement that $\text{limsup}~\mu_n(F)\leq\mu(F)$ for all closed sets $F$. In your question the set $(-\infty,x]$ is closed in $\mathbb{R}$. Hence, there is no counterexample.