Normal Random Variables and convergence in distribution.

Question

I'm stuck with a problem of probability. Suppose $\{N_n,n \geq 0\}$ is a sequence of normal random variables. Show $N_n \Rightarrow N_0$ iff $$E(N_n) \to E(N_0)$$ and $$Var(N_n) \to Var(N_0).$$ I'm done when I suppose the convergence of the expected value and the variance, but i'm not able to make a proof when I suppose $N_n \Rightarrow N_0$.

Answer 1

Weak convergence is equivalent to convergence of characteristic functions. Take absolute value in characteristic functions to see that the variances converge. Then it becomes obvious that the means also converge.