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I am familiar with another answer about this topic: How to prove Poisson Distribution is the approximation of Binomial Distribution? , but i don't understand one part of it.

Could anyone explain to me the following part;

For $i$ fixed and $n$ large we have $$ \frac{n!}{n^i \cdot (n-i)!} = \frac{n(n-1)(n-2)\cdot \ldots \cdot (n-i+1) }{ n^i} \approx 1 $$

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$$\require{cancel}\dfrac{n!}{n^i(n-i)!}=\dfrac{\cancel{1\cdot 2\cdot 3\cdot\ldots\cdot(n-i)}\cdot\overbrace{(n-i+1)\cdot\ldots\cdot (n-2)\cdot (n-1)\cdot n}^i}{\cancel{1\cdot 2\cdot 3\cdot\ldots\cdot(n-i)}\cdot \underbrace{n\cdot n\cdot\ldots\cdot n}_i}=$$ $$=\dfrac{n}{n}\cdot \dfrac{n-1}{n}\cdot\dfrac{n-2}n\cdot\ldots\cdot\dfrac{n-i+1}{n}\to 1 \text{ as } n\to\infty.$$ Last expression contains the product of $i$ fractions. Each of these fractions converges to $1$ as $n\to\infty$. Number of factors is fixed, so the limit of a product equals to the product of limits.

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    +1 The bit about "numbers of factors is fixed" is essential; otherwise one could wrongly conclude that $n!/n^n \to 1$ – leonbloy Mar 09 '17 at 14:33