Let $(X_1,...,X_n)$ an exchangeable family of real random variables such that its empirical measure $\mu_n:=\frac{1}{n} \sum_{k=1}^n \delta_{X_k}$ converges to some probability measure $\mu$. The convergence is in probability, that means: for any $f$ bounded continuous, $\frac{1}{n} \sum_{k=1}^n f(X_k)$ converges in probability to $\int f d\mu$.
I would like to prove that $X_n$ converges in distribution to $\mu$. I don't know whether this fact is true or not. Any help would be appreciated.