Limit in distribution of increasing variance normal random variable

Question

Suppose you have a sequence of normal random variables $X_n \sim N(0, n)$. Is there any random variable $X$ such that $X_n \to X$ in distribution.

Here, I use the following definition: $X_n \to X$ in distribution if $P(X_n \leq x) \to P(X \leq x) \equiv F_X(x)$ for every $x$ for which $F_x$ is continuous.

Answer 1

$X_n \overset{d}{=} \sqrt{n} Z$ where $Z \sim N(0, 1)$ so $$P(X_n \le x) = P(Z \le x / \sqrt{n}) \to \frac{1}{2}.$$ Now show that no CDF can equal $1/2$ wherever it is continuous.