= Solution
{c}
There is a stronger conclusion than the requested limit. For $x>0$, the event $N_n-n\geq x\sqrt n$ forces $N_n\geq n+1$, 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 $T_n\leq n$. This contradicts $T_n-n\geq y\sqrt n$ for $y>0$. Thus the two events are disjoint:
$$
\boxed{\mathbb P\!\left(\frac{N_n-n}{\sqrt n}\geq x,\frac{T_n-n}{\sqrt n}\geq y\right)=0\quad\text{for every }n.}
$$
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