stochastic processes - Derive Poisson distribution from probability per time of event

Suppose we have a probability per time $\lambda$ that something (e.g. nuclear decay, random walk takes a step, etc.) happens. It is a known result that the probability that $n$ events happen in a time interval of length $T$, is given by the Poisson distribution $$P(n) = \frac{e^{-\lambda T} (\lambda T)^n}{n!} \, .$$ How do we prove this?