For problem 4 of homework 2, it would be sufficient to prove that $\langle n^{(0)} | {\underline n}^{(j+1)} \rangle = 0$ for any $j = 0, 1, 2, ...$. The ket $| {\underline n}^{(j+1)} \rangle$ is defined in this image. In the same image, you can find other information. For example, four completely equivalent statements, important for this problem, are noted in the green rectangle. The equivalence of these four statements can be proven based on Eqs. 3.19 and 3.20, alone. This is kind of demonstrated in this image, and you are not required to demonstrate it yourself. Just prove that $\langle n^{(0)} | {\underline n}^{(j+1)} \rangle = 0$ for any $j = 0, 1, 2, ...$ (using the proof by induction).—3:35PM, Oct 14, 2013