# Normal Random Variables and convergence in distribution.

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 $$N_n \Rightarrow N_0$$.

• Can you clarify your comment? Since the normal distribution is entirely characterized by its mean and variance, and convergence in distribution is a pretty weak convergence, it almost seems self-evident that this is true. What did you mean by "I'm not able to make a proof..." – eSurfsnake May 28 '19 at 0:34

• $e^{itx}$ is a bounded continuous function so the characteristic functions converge. – Kavi Rama Murthy May 28 '19 at 11:33