Consider a Poisson process with intensity function $\{\lambda(t),t\ge 0\}$. How should I prove that the number of arrivals in interval $[t,t+s]$ follows a Poisson distribution as follows: $$ P\{ N(t+s)-N(t)=n\}=e^{-[m(t+s)-m(t)]} \frac{\big[m(t+s)-m(t) \big]^n}{n!}, \quad n=0,1,... $$ where $$ m(t)=\int_{0}^{t}\lambda(u)du. $$
I wanna prove it by induction, but I failed to do it.