Are these two statements about the probability tending to Normal distribution equivalent?

Question

Are these two statements equivalent:

1. $P(X_n < \beta) \to z(\beta)$

2. For all fixed $\alpha \text{ < } \beta \in \mathbb{R} \quad P(\alpha < X_n < \beta) \to z(\beta) - z(\alpha)$

$z(\beta) = P(X < \beta)$ where $X$ ~ $N(0,1)$

I can see that the first implies the second one. But I'm not sure if the second implies the first. I think we might be able to use Fatou's lemma here.

Answer 1

The answer is YES. Let $F(x)=P(X\leq x),F_n(x)=P(X_n \leq x)$ and $\epsilon >0$. There exists $M$ such that $1-F(M)+F(-M) <\epsilon$. Hence there exists $n_0$ such that $1-F_n(M)+F_n(-M) <\epsilon$ for all $n \geq n_0$. For any $x$ we have $F_n(x)=[F_n(x)-F_n(-M)]+F_n(-M)$ and $F(x)=[F(x)-F(-M)]+F(-M)$. Note that $F_n(-M)<\epsilon$ and $F(-M)<\epsilon$. Can you finish?