Limit of Poisson random variables with parameters tending to infinity

Question

I have two independent Poisson distributed random variables: $\xi$ with parameter $\lambda$ and $\eta$ with parameter $\mu$. I need to show that $${{(\xi - \lambda) - (\eta - \mu)}\over{\sqrt{\xi + \eta}}} \rightarrow N(0,1)\:as\:\lambda,\mu\:\rightarrow\infty $$ where $N(0,1)$ is standard normal distribution. I've tried to do it using the fact that Poisson distribution with parameter $\lambda$ can be approximated by Normal when $\lambda$ is large enough, but i am confused with calculatng the ratio of random variables.
Will be very glad for any hints.