The problem and its solution are given below:
But I do not understand the idea of the solution and some detailed steps , like why we take $\mathcal{F}$ in this form and why the absolute continuity definition assures the existence of such $N$ and why in the last line the indices of the summation one time starts from $j=o$ while the other time starts from $j=1$ and why they end at the same $N$ could anyone explain it for me please?