Fix $\lambda>0$, and let $p_{\lambda}:\mathbb{N}\to\mathbb{R}$ be the probability mass function (pmf) of Poisson distribution: \begin{align} p_{\lambda}(k)=\frac{\lambda^ke^{-\lambda}}{k!},\qquad\forall k\in\mathbb{N} \end{align} It is well known that Poisson distribution can be derived as the limit of Binomial distribution (with parameters $n$ and $p$) as $n$ becomes large and $np$ approaches a finite limit, a fact known as the Poisson limit theorem or the Law of rare events. One version of this says that with $np=\lambda$, the binomial distribution converges to the Poisson distribution as $n\to\infty$. More specifically, denote the pmf of binomial distribution with parameter $n$ by $p_n:\mathbb{N}\to\mathbb{R}$, given by \begin{align} p_n(k)&=\begin{pmatrix}n \\ k\end{pmatrix}p^k(1-p)^{n-k} \\ &=\frac{n!}{k!(n-k)!}\left(\frac{\lambda}{n}\right)^k\left(1-\frac{\lambda}{n}\right)^{n-k},\qquad\forall 0\leq k\leq n \end{align} (and $p_n(k)=0$ for all $k>n$). Then the theorem says that $\forall k\in\mathbb{N}$, \begin{align} \lim_{n\to\infty}p_n(k)=p_{\lambda}(k) \end{align}
In this post I am concerned with the question of whether the convergence is uniform. In terms of the notations set up above:
Is it true that $p_n$ converges to $p_{\lambda}$ uniformly on the whole domain $\mathbb{N}$ as $n\to\infty$? That is, does it hold that \begin{align} \lim_{n\to\infty}\sup_{k\in\mathbb{N}}\left|p_n(k)-p_{\lambda}(k)\right|=0? \end{align}
A simply google search (with keywords like binomial, Poisson, uniform convergence) seems to be futile to get something useful, perhaps this question is not that interesting from the point of view of probability theory? Or maybe I searched it without the correct keywords? In any case, it seems a natural question to ask from the simple point of view of analysis.
Any comment, suggestion or answer are welcome and appreciated.
Yes, the uniform convergence holds. In fact, a stronger conclusion is true: we have convergence in total variation distance, i.e. for $n \to \infty$ $$ \sum_{k=0}^{\infty} |p_n(k) - p_{\lambda}(k)| \to 0. $$ This is a consequence of Le Cam's theorem.