Let $\{X_n\}$ be such that $X_n$ has a binomial distribution with parameters $n$ and $p=\lambda /n$ , then as known $X_n$ will converge in distribution to $Y$ which has a Poisson distribution with parameters $\lambda=n.p$ .
Now if we define a sequence $\{Z_n\}$ by $Z_n =g(X_n,n)=X_n/n$ then the Moment generating function of $Z_n$ will be $$M_{Z_n}(t)=(1+(\lambda/n)(exp(t/n)-1))^n$$ then as $n \rightarrow \infty$ , $M_{Z_n}(t) \rightarrow 1 $
My question is , is that mean that $P(z=1)=1$ or $P(z=0)=1$ ?