Convergence in Distribution Suppose $X_n$ are $\mathbb{R}$-valued random variables. Then how would I be able to show that $X_n\rightarrow^D X$ where $X$ is also a $\mathbb{R}$-valued random variable iff $F_n(x)\rightarrow F(x)$ for all $x\in A$ where $A$ is a dense subset of $\mathbb{R}$ and $F_n$ and $F$ are respective distribution functions. 
 A: We want to show that:
$$ \lim_{n\rightarrow \infty} F_n(x) = F(x), \ \forall x \in C_F \Longleftrightarrow  \lim_{n\rightarrow \infty} F_n(x) = F(x), \ \forall x \in D \subseteq \mathbb{R}, \text{ D is dense}$$
To show $\Rightarrow$, let $D=C_F$. We have proven earlier in the course that for any distribution function $F$, there are at most countable discontinuities. For any set $A \subseteq \mathbb{R}$, if $A^C$ is countable, then $A$ is dense in $\mathbb{R}$. By assumption, $\lim_{n\rightarrow \infty} F_n(x) = F(x), \ \forall x \in C_F = D$ so the first direction is proven. 
To show $\Leftarrow$, we assume that $\exists D \subseteq \mathbb{R}$ where $D$ is dense in $\mathbb{R}$ such that $\lim_{n\rightarrow \infty} F_n(x) = F(x), \ \forall x \in D$. Take any $x \in C_F$ we want to look at sequences $l_i \subset D,\ u_i \subset D $ such that $l_i \uparrow x$ and $u_i \downarrow x$. Because $F_n$ is a nondecreasing function, $l_i \le x \le u_i \Rightarrow F_n(l_i) \le F_n(x) \le F_n(u_i)$. By assumption, we have $ \lim_{n\rightarrow \infty} F_n(l_i) = F(l_i)$ and $ \lim_{n\rightarrow \infty} F_n(u_i) = F(u_i)$. So, as $n\rightarrow \infty$, $ F(l_i) \le F_n(x) \le F(u_i)$. Because $x$ is a continuity point of $F$, we have $\lim_{y\rightarrow x} F(y) = F(x)$. Then $\lim_{i\rightarrow \infty} F(u_i) = F(x)$ and $\lim_{i\rightarrow \infty} F(l_i) = F(x)$. This means that as $i \rightarrow \infty$, $ F(x) \le F_n(x) \le F(x)$. So for $n \rightarrow \infty, \ i \rightarrow \infty$ we have $$\lim_{n\rightarrow\infty,\ i\rightarrow \infty}F_n(x) = \lim_{n\rightarrow\infty}F_n(x) = F(x)$$ Since this is true for any $x \in C_F$ we have $X_n \rightarrow^D X$.
Reference:
http://cmi.ac.in/~vinay/notes/measure-final.pdf
