Show that $P(X_n \leq x_n) \rightarrow P(X\leq x)$

Question

Suppose $X_n$ converges in distribution to $X$, $x_n\rightarrow x$ and the cumulative distribution function for $X$ is continuous at $x$. Show that $P(X_n \leq x_n) \rightarrow P(X\leq x)$

Answer 1

Hint Remember the Kolmogorov's Axioms of Probability and definition convergence of sequence of sets type $[X\leq x_n]$.

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