Does pointwise convergence of CDFs imply convergence in distribution?

Question

Casella & Berger define convergence in distribution as follows:

$X_n \rightarrow X $ (in distribution) if $$\lim_{n\rightarrow \infty} F_{X_n}(x)=F_X(x)$$ for all $x$ such that $F_X(x)$ is continuous.

This resembles pretty much the concept of pointwise convergence of a sequence of functions, so is it fair to say that if $\{F_{X_n}\}$ converges pointwise to $F_X$ then $\{X_n\}$ converges in distribution to $X$? (the converse may not be true since the convergence in distribution requires only convergence in the continuity points). Is my reasoning ok?

Thanks in advance.

Answer 1

Pointwise convergence gives convergence at every point. In particular we have convergence at continuity points so we surely have $X_n\to X$ in distribution. To show that the converse is not true take $X_n =\frac 1 n, X=0$ and consider convergence at $0$.