This is an exercise of the Central Limit Theorem:
Let $Y^{\lambda}$ be a Poisson random variable with parameter $\lambda>0$. Prove that $\frac{Y^{\lambda}-\lambda}{\sqrt{\lambda}}\to Z\sim N(0,1)$ in distribution as $\lambda\to\infty$.
I've done that $ Z_n\to Z\sim N(0,1) $ in distribution using the CLT, where $Z_n=(Y^n-n)/\sqrt{n}$. Some naive attempt to go is considering $ Y^{n}\leq Y^{\lambda}\leq Y^{n+1}\tag{*} $ where $n\leq\lambda\leq n+1$ and somehow use the squeeze theorem. But both (*) and the squeeze theorem in convergence in distribution are NOT justified. How can I go on? Or do I need an alternative direction?