Convergence in distribution of random variables question $X, X_1, X_2,\ldots $ are real random variables with $\mathbb{P}(X_n\leq x)\to \mathbb{P}(X\leq x)$ whenever $\mathbb{P}(X=x)=0$.
Why does $X_n\stackrel{L}{\to} X$? At the least, where would I begin?

A sequence $X_1,X_2,\ldots$ of random variables is said to converge in distribution, or converge weakly, or converge in law to a random variable $X$ if
$$
\lim_{n\to\infty}F_n(x)=F(x)
$$
for every number $x\in\mathbb R$ at which $F$ is continuous, where $F_n(x)=\mathbb P(X_n\le x)$ and $F(x)=\mathbb P(X\le x)$.
Thus, we need to show that $F(x)$ is continuous at $x$ if and only if $\mathbb P(X=x)=0$. $F$ is continuous from the right, so we need to investigate continuity from the left. Since 
$$
\mathbb P(X=x)=F(x)-\lim_{y\uparrow x}F(y),
$$
we have that $\mathbb P(X=x)=0$ if and only if $F(x)=\lim_{y\uparrow x}F(y)$.
