Binomial Distribution Approximated by Poisson Distribution

Theorem
Let $X$ be a discrete random variable which has the binomial distribution with parameters $n$ and $p$.

Suppose $n$ is "very large" and $p$ is "very small", but $np$ of a "reasonable size".

Then $X$ can be approximated by a Poisson distribution with parameter $\lambda$ where $\lambda = np$.

Proof
Let $X$ be as described.

Let $k \ge 0$ be fixed.

We write $p = \dfrac \lambda n$ and suppose that $n$ is large.

Then:

Hence the result.

Comment
Okay wise guy, exactly what constitutes "very large", "very small", and "of a reasonable size"?

Well, if $n = 10^6$ and $p = 10^{-5}$, we have $np = 10$.

That's the sort of order of magnitude we're talking about here.