Convergence in distribution equaling a certain value

Question

Does the probability of something converging in distribution to N(0,1) equaling a certain value converge to 0?

$\mathbb{P}(X = a) \xrightarrow[]{} 0 \text{, where }X \xrightarrow[]{D} N(0,1), a\in \mathbb{R}$

Intuitively this is true as X is more and more continous and for continous random variables the probability is 0, but I don’t know how to show it.

Answer 1

Yes. Since $\sup_{\{x \in \mathbb R\}} |P(X\leq x) -\Phi(x)| \to 0$ we get $\sup_k |P(X \leq a-\frac 1 k)-\Phi(a-\frac 1 k)| \to $ and this gives $|P(X < a)-\Phi(a)| \to 0$. Hence $P(X=a)=P(X \leq a)-P(X<a) \to 0$.