Probability – Show that the number of elements in a given set is a Poisson process.

To let $ quad { nu_n, xi_k ^ n } in mathbb N $ to be a family of random variables. $ quad nu_n sim Poiss ( lambda), quad xi_k ^ n sim Unif (n-1, n], $
To let $ quad N_t = | {(k, n): k le nu_n, quad xi_k ^ n le t } |. $
I have to show that $ N_t $ is Poisson process on $[0+infty)$[0+infty)$[0+infty)$[0+infty)$ and I have trouble proving the definition (I use the definition of the Poisson process as the Levy process). For hints, I would be very thankful.