How to prove Poisson Distribution is the approximation of Binomial Distribution?

I was reading Introduction to Probability Models 11th Edition and saw this proof of why Poisson Distribution is the approximation of Binomial Distribution when n is large and p is small:

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I can understand most part of the proof except for this equation:
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I really don’t remember where it comes from, could anybody explain this to me? Thanks!.

Solutions Collecting From Web of "How to prove Poisson Distribution is the approximation of Binomial Distribution?"

Well, this is a basic fact of the exponential function $e^x$.

One definition of $e$ is the limit $\lim_{n\to\infty}(1+\frac1n)^n$. By a monotonicity argument one can prove $\lim_{x\to\infty}(1+\frac1x)^x=e$ where $x$ now ranges the real numbers.

Also note that $1-\frac1x=\frac{x-1}x=1/\frac x{x-1}=1/(1+\frac1y)$ where $y=x-1$.

So, $\lim_{x\to\infty}(1-\frac1x)^x=\lim_{y\to\infty}(1+\frac1y)^{-(y+1)}=e^{-1}$ (using $(1+\frac1y)\overset{y\to\infty}\to 1$ for the extra $+1$ in the exponent).

From here, assuming $\lambda>0$,
$$\begin{aligned}e^{-\lambda}=(e^{-1})^y &= \lim_{x\to\infty}(1-\frac1x)^{\lambda x} =&\to_{\ z:=\lambda x}
\\
&= \lim_{z\to\infty}(1-\frac\lambda z)^z\,.
\end{aligned} $$

In consequence, we have this limit for every sequence $z_n\to\infty$ written in place of $z$ and limiting on the natural $n\to\infty$. In particular, this also holds for $z_n=n$.

Note that we had to take the turnaround for arbitrary real numbers instead of integers only because of the exponent $\lambda$.