I want to prove that a Poisson process fulfils the continuity in probability property (https://en.wikipedia.org/wiki/L%C3%A9vy_process#Mathematical_definition)
$lim_{h \rightarrow 0} \mathbb{P}(|N_{t+h}-N_{t}|> \epsilon) = 0$ whereas $N_t=\sum_{n \geq 1} \mathbb{1}_{(0,t]} T_n$ ($t \geq 0$ Tn are the jumps)
so what I have: $N_{t+h}-N_t =\sum_{n \geq 1} \mathbb{1}_{(0,t+h]} T_n- \sum_{n \geq 1} \mathbb{1}_{(0,t]} T_n= \sum_{n \geq 1} \mathbb{1}_{[t,t+h]} T_n$
so it would be clear as $h \rightarrow 0$ then $N_{t+h}-N_t$ would equal zero, however how to prove it rigorously with the $\epsilon$ definition and how could I justify interchanging limit and probability?