Let $P$ be a Poisson process with parameter $\lambda$. The goal is to show that $$ \lim _{n \rightarrow \infty} \sup _{t \leq 1}\left|\frac{P_{n t}}{n}-\lambda t\right|=0, \quad \text { a.s. } $$
This seems like an easy application of the Borel-Cantelli. Let $A_n = \{\vert\frac{P_{nt}}{n} - \lambda t\vert > \epsilon \text{ for some } t \in [0,1]\}$, we just need to bound $P(A_n)$. However, I am having trouble dealing with the supremum over $t$ here.
Any hint? Solution is also welcome.