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 $$p = \frac \lambda n$$ 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.