Let $(P(t))_{t\geq0}$ be a Poisson process of parameter $1$ (for instance, let $P(t) = \mathrm{Card}\{k \in \mathbb{N}^*: S_k \leq t \}$ where $S_k = E_1+\ldots+E_k$ is a sum of $k$ i.i.d exponential of mean $1$).
I showed that for $T>0$ fixed, almost surely, as $n\to\infty$, $$ \sup_{t\in [0,T]} \left| \frac{P(nt)}{n}-t \right| \to 0.$$ I did that following very closely the proof of the Glivenko-Cantelli theorem, using that $P(t)$ is increasing in $t$ and Dini's second theorem.
I am trying to prove that this is not true for $T = \infty$. I don't really know how to tackle that problem. The only step which doesn't work when $T=\infty$ is using Dini's theorem, which requires compactness (for instance $f_n(t) = t/n$ is a counterexample which doesn't work without compactness).
Let $$ Y_n:=\sup_{t\geqslant 0}\left\lvert\frac{P(nt)}{n}-t\right\rvert. $$ Doing the change of variable $y=nt$ for a fixed $n$, one has $$ Y_n:=\frac 1n\sup_{y\geqslant 0}\left\lvert P(y) -y\right\rvert. $$ Therefore, it suffices to prove that $Y:=\sup_{y\geqslant 0}\left\lvert P(y) -y\right\rvert$ is not almost surely finite. If $Y$ was almost surely finite, then so would be $Z:=\sup_{y\geqslant 0}\left\lvert P(y+1)-P(y)\right\rvert$. Using the second Borel-Cantelli lemma (applied the events $A_y:=\{\left\lvert P(y+1)-P(y)\right\rvert\geqslant k_0\}$, $y\in\mathbb N$), we can see that $Z\geqslant k_0$ almost surely. In conclusion, $Y_n$ is almost surely infinite.