CLT for sequences

Question

Assume $\forall y\in\mathbb{R}$, $P(Y_n\leq y)\rightarrow\Phi(y)$ and if $n\rightarrow\infty$, $x_n\rightarrow x$, prove $P(Y_n\leq x_n)\rightarrow\Phi(x)$ if $x_n\rightarrow\infty$. It seems very intuitive, but I have no idea how to prove it.

Answer 1

In your case, the normal distribution function $\Phi(x)$ is a continuous function, hence by Polya's theorem, $\mathsf{P}(Y_n\le y)\to \Phi(y)$ uniformly in $y$, that is $$ \lim_{n\to \infty}\sup_{y\in\mathbb{R}}|\mathsf{P}(Y_n\le y)- \Phi(y)|=0. $$ Therefore, \begin{align} |\mathsf{P}(Y_n\le x_n)- \Phi(x)|&\le |\mathsf{P}(Y_n\le x_n)-\Phi(x_n)|+|\Phi(x_n)-\Phi(x)|\\ &\le \sup_{y\in\mathbb{R}}|\mathsf{P}(Y_n\le y)- \Phi(y)|+|\Phi(x_n)-\Phi(x)|.\tag{1} \end{align} From (1) it is easy to get what you want.