= Solution
For fixed $k$ and sufficiently large $N$,
$$
p_k(N,\lambda/N)
=\frac{\lambda^k}{k!}\left[\prod_{j=0}^{k-1}\left(1-\frac jN\right)\right]
\left(1-\frac\lambda N\right)^N
\left(1-\frac\lambda N\right)^{-k}.
$$
The finite product and last factor tend to one, while $(1-\lambda/N)^N\to e^{-\lambda}$. Thus the <Poisson limit theorem> gives
$$
\boxed{\lim_{N\to\infty}p_k(N,\lambda/N)=e^{-\lambda}\frac{\lambda^k}{k!}.}
$$
These numbers form a <probability mass function> because they are nonnegative and their sum is $e^{-\lambda}\sum_{k\geq0}\lambda^k/k!=1$, by the <exponential series>. They define the <Poisson distribution> of parameter $\lambda$. Only $N\geq\lambda$ is needed to make the binomial parameter valid, which is sufficient for the limit.
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