Given a probability space, $(\Omega, \mathcal{F}, P)$, how to show that $\{P(A): A \in \mathcal{F}\}$ is closed?

Question

Given a probability space, $(\Omega, \mathcal{F}, P)$, how to show that $\{P(A): A \in \mathcal{F}\}$ is closed?

I tried showing this by using the fact that a set is closed if and only if each limit point is contained in it. But the convergence of $\{P(A_n)\}$ seems to imply nothing about the property of the sets $\{A_n\}$.

I also tried proving this by dividing it into two distinct cases:

1. $(\Omega, \mathcal{F}, P)$ is nonatomic. In this case, $\{P(A): A \in \mathcal{F}\}$ is closed.
2. $(\Omega, \mathcal{F}, P)$ is atomic. But in the second case, things are still complicated.