Let $~\{ \nu_n, \xi_k^n \}\in \mathbb N$ be a family of random variables. where $~ \nu_n \sim \mathrm{Pois}(\lambda)$ and $~\xi_k^n \sim \mathrm{Unif}(n-1,n].$
Consider $~ N_t=\bigl| \{ (k,n): k \le \nu_n, \quad \xi_k^n\le t \} \bigr|.$
I need to show that $ N_t$ is Poisson process on $[0, +\infty)$ and I have problems proving the definition (I use definition of Poisson process as a Levy process). I would be grateful for any advice.