We will talk about weak convergence in general metric spaces in Chapters 30–35. This exercise is concerned with the weak convergence of real-valued random variables as defined in Section A.12. Suppose...

We will talk about weak convergence in general metric spaces in Chapters 30–35. This exercise is concerned with the weak convergence of real-valued random variables as defined in Section A.12.

Suppose for each n, Pn is a Poisson random variable with parameter
  as n → ∞. Prove that

converges weakly to a normal random variable with mean zero and variance one.