I would like to see which steps are required in order to solve the problem.
How to write N=N_{1 }U N_{2 }U N_{3}.... U N_{n}U.... as an infinite union of infinite subsets pairwise disjoint
True, given an infinite union of sets:
$N = \bigcup_{k\ge 1}U_k$, and you want to find an infinite union of pairwise disjoint sets, define $A_n = \bigcup_{k=1}^n U_k$ Then define $N_1 = U_1$ and $N_{n+1} = U_{n+1} \setminus A_n$ for all $n>0$. Can you show that $N = \bigcup_{k\ge 1}N_k$? Can you also show that the $N_k$'s are pairwise disjoint?