I need a check on this procedure.
Let $(N_t)_t$ be a homogeneous poisson process. Show that $$N_s \rightarrow N_t$$, for $s<t$ in probability as $s \rightarrow t^-$.
Let $\eta>0$. Consider the set $\{ w \in \Omega: |N_t - N_s| > \eta\}$ and recall that $N_t - N_s $ is distributed like $ \text{Poiss}(\theta(t-s))$.
Now, $$P(|N_t - N_s|>\eta) \leq \frac{E[|N_t-N_s|]}{\eta} = \frac{E[N_t - N_s ]}{\eta} = \frac{\theta(t-s)}{\eta}$$
THen, by letting $s \rightarrow t^{-}$, I have that the right hand side goes to $0$, and hence convergene in probability is proved.
Is everything okay?