I'm reading a book and this proof/derivation is inside on the first page explaining the Poisson distribution. However, there is part of it i do not understand if someone could help me that would be great. They are deriving it from the binomial distribution. So to start they say
Letting $\lambda = np$ and taking the limit of the binomial probability $p(y) = {n\choose y}p^y(1-p)^{n-y}$ as $n \rightarrow \infty$ we have
$$\lim_{n \to \infty }{n \choose y}p^y(1-p)^{n-y}=\lim_{n \to \infty}\frac{n(n-1)...(n-y+1)}{y!}\bigg(\frac{\lambda}{n}\bigg)^y\bigg(1 - \frac{\lambda}{n}\bigg)^{n-y}$$ $$=\lim_{n \to \infty}\frac{\lambda^y}{y!}\bigg(1-\frac{\lambda}{n}\bigg)^n\Bigg[\frac{n(n-1)...(n-y+1)}{n^y}\Bigg]\bigg(1 - \frac{\lambda}{n}\bigg)^{-y}$$ $$=\frac{\lambda^y}{y!}\lim_{n \to \infty}\bigg(1-\frac{\lambda}{n}\bigg)^n\bigg(1 - \frac{\lambda}{n}\bigg)^{-y}\Bigg[\bigg(1-\frac{1}{n}\bigg) \times \bigg(1-\frac{2}{n}\bigg) \times ...\times\bigg(1-\frac{y-1}{n}\bigg)\Bigg]$$Noting that $$\lim_{n \to \infty}\bigg(1-\frac{\lambda}{n}\bigg)^n=e^{-\lambda}$$ and all other terms to the right of the limit have a limit of 1, we obtain $$p(y) = \frac{\lambda^y}{y!}e^{-\lambda}$$
So the part that I don't understand is the transformation in the square brackets. How do you get from the square brackets in line 2 to the brackets in line 3 please?
Adding a couple more steps: $$\begin{align}\frac{n(n-1)\cdots(n-y+1)}{n^y} &=\frac{\overbrace{n(n-1)\cdots(n-y+1)}^{y\text{ of these}}}{\underbrace{n\cdot n\cdots n}_{y\text{ of these}}}\\ &= \underbrace{\frac{n}{n}\cdot\frac{n-1}{n}\cdots \frac{n-y+1}{n}}_{y \text{ of these}} = \\&=1\cdot\left(1-\frac1n\right)\left(1-\frac2n\right)\cdots \left(1-\frac{y-1}n\right)\end{align}$$
Is any step left unclear now?