There's an exercise in a book I'm reading and I'm a little unsure of how to do it with how it is written. It goes:
Let $Z_n\xrightarrow{dist}Z$ where $Z\sim F(x)$. If $x_n\rightarrow x$, a sequence of real numbers converging to $x$ a point of continuity of $F$, then $\displaystyle\lim_{n\to\infty}P(Z_n\leq x_n)=P(Z\leq x)=F(x).$
My first thought was continuity of probability, but $x_n$ isn't necessarily monotonic. I think this comes together pretty easy if you take the indices under two separate limits, but that is not how this is written and I'm not sure if that involves Monotone Convergence theorem to justify.