Suppose $\{P_n\}$ and P are probability measures on the real line with corresponding distribution functions $\{F_n\}$ and $F$, respectively.
Prove that $P_n$ converges weakly to P if and only $$\lim_{n \rightarrow \infty} F_n(x) = F(x)$$ at every point x where F is continuous.
We define weak convergence as: Let S be a metric space with its Borel $\sigma$-algebra $\Sigma$. We say that a bounded sequence of positive probability measures $P_n$ on $(S, \Sigma)$, $n = 1, 2, ...,$ converges weakly to the finite positive measure P, and write: $P_n \Rightarrow P$
Could we somehow use the fact that the probability measure for an interval $(a,b)$ = $F(b)-F(a)$ to answer this question?