Counting random exponential events

Question

Let's say we have random events happening with the exponential probability: $P(t_i>T)=e^{-\lambda T}$, where $i=1,..,n$. What is the probability that there will be $k$ events within time period $(0,T)$? My understanding is that it should be a Poisson distribution with mean $\lambda$, but why and how can I prove it rigorously?