Suppose $X$ is a Poisson$(\lambda)$ random variable. I have already shown as part of the question that $\sqrt{\lambda}\Pr(X=\lambda+x\lambda) = \frac{1}{\sqrt{2\pi}}\exp^{\frac{-x^2}{2}}$ but I need to show that $\Pr(a < \frac{X-\lambda}{\sqrt{\lambda}}<b)$ tends to $\int_{b}^{a}\varphi(w)\,dw$ as $\lambda \rightarrow \infty$. I have realised that the mean is $\lambda$ and Standard deviation $\sqrt{\lambda}$.
I thought that all that was necessary was to apply the central limit theorem but I feel that it is more complicated than that. Any help would be much appreciated.