A weaker characterization of convergence in distribution $X_n$ and $X$ are real-valued random variables. If $X_n \to X$ in distribution, then we know that
$$P\{X_n \leq a\} \to P\{X \leq a\}$$
for every $a$ at which $x\mapsto P\{X \leq x\}$ is continuous. For a certain problem I have at hand this condition is too strong to check. Instead, I only have the following.
$$P\{X_n \leq a\} \to P\{X \leq a\}$$
for every $a \in D$ where $D = \{d_1,d_2,\ldots\}$ is a countable dense set in $\mathbb{R}$ such that $x\mapsto P\{X \leq x\}$, $x\mapsto P\{X_1 \leq x\}$, $x\mapsto P\{X_2 \leq x\},\ldots$ are continuous at $d_i$ for every $i = 1,2,\ldots$. I would like to show that this condition is sufficient for convergence in distribution.
I fix an arbitrary $d \notin D$ such that $x\mapsto P\{X \leq x\}$ is continuous at $d$. By denseness of $D$ I can find a sequence $(d_k)_k$ such that $d_k \downarrow d$. By right-continuity and monotonicity of the distribution functions
$$P\{X_n \leq d_k\} \to P\{X_n \leq d\} \quad \text{and} \quad P\{X \leq d_k\} \to P\{X \leq d\}$$
for every $n$ as $k \to \infty$. Furthermore, by definition of the set $D$
$$P\{X_n \leq d_k\} \to P\{X \leq d_k\}$$
for every $k$ as $n\to\infty$.
How do I conclude rigorously from these that
$$P\{X_n \leq d\} \to P\{X \leq d\}$$

If a sequence $(d_k)$ from $D$ decreases to $x$, then $F_n(x)\le F_n(d_k)$, so $\limsup_nF_n(x)\le F(d_k)$ for all $k$, so $\limsup_nF_n(x)\le F(x)$. If $(d_k)\subset D$ increases to $x$, and $x$ is a continuity point of $F$, then in like fashion $\liminf_nF_n(x)\ge F(x)$. 
