I'm trying to show that for a Poisson random variable $\displaystyle Y$ with mean $\displaystyle \lambda$, we get

$\displaystyle \displaystyle\frac{Y-\lambda}{\sqrt{\lambda}}\Rightarrow N$

as $\displaystyle \lambda\rightarrow\infty$

If $\displaystyle \lambda$ is a natural number n I'm okay, since $\displaystyle Y$ is the sum of n independent Poisson variables with mean 1, and the CLT applies.

My problem is extending to $\displaystyle \lambda$ not necessarily a natural number. Any advice? Thanks!