Past exam of the probability course of the University of Oxford / 2021 / 1 / c / iii / Solution

There is a stronger conclusion than the requested limit. For $x>0$, the event $N_n-n\ge x\sqrt{n}$ forces $N_n\ge n+1$, because $N_n$ is an integer. In particular, at least $n$ arrivals have occurred by time $n$, so the $n$-th Poisson arrival time satisfies $T_n\le n$. This contradicts $T_n-n\ge y\sqrt{n}$ for $y>0$. Thus the two events are disjoint: