[Math] Weak convergence of Dirac measures converges to a Dirac measure

Question

Let $X$ be a metrizable space and $\{x_n\}$ be a sequence in $X$. Suppose the sequence of Dirac measures $\delta_{x_n} \xrightarrow{w} P$ where $P$ is some probability measure. Prove that $P = \delta_x$ for some $x \in X$.

I am not sure how to go about proving this exactly…

The definition of weak convergence I am using is the standard one: so $\delta_{x_n} \xrightarrow{w} P$ happens when for every continuous and bounded $f: X \xrightarrow{} \mathbb{R}$ we have $\int f d\delta_{x_n} \xrightarrow{} \int f dP$.

Answer 1

We can follow (and detail) the following steps.

1. Fix an integer $r$ and construct a continuous and bounded function $f$ such that $f(x)=1$ if $x$ belongs to the closure of $F_r:=\bigcap_{n\geqslant r}\{x_n\}$.
2. The definition of weak convergence implies that for each $r$, $\mathbb P(\overline{F_r})=1$.
3. In particular, $\bigcap_r\overline{F_r}$ is non empty. It remains to prove that this set cannot contain two distinct elements