Convergence in distribution of $N(0, 1/n)$

I am a graduate student in statistics and am self-studying convergence in probability. I am a little confused on the following problem.

I am trying to follow a proof that claims
$$X_nsim N(0,1/n) overset{D}{rightarrow} 0.$$

The proof uses the definition
$$lim_{n to infty} F_n(t) = F(t)$$
to show the convergence in distribution, where $F(t)$ is c.d.f. of the point mass distribution at $0.$ The proof shows that, for $t<0$,
$$lim_{n to infty} F_n(t) = mathbb{P}(X_n < t) = 0,$$
and that for $t>0$,
$$lim_{n to infty} F_n(t) = mathbb{P}(X_n < t) =1.$$
The proof then concludes that $X_n overset{D}{rightarrow} 0.$

My question:

I completely understand the steps of the proof and how the limits were found, which is why I chose not to include them here. I do not understand the last statement. Wouldn’t it be more correct to say that $X_n overset{D}{rightarrow} X$, where $X$ is the point mass distribution at $0$? The definition of convergence of distribution states that a random variable converges to another random variable. $0$ is not a random variable.

Thank you.