For nucleons we have that the maximum value of \( M_S=m_s=1/2 \), yielding $$ (m_j)_{\mathrm{max}}=l+\frac{1}{2}. $$ Using this and the fact that the maximum value of \( M_J=m_j \) is \( j \) we have $$ j=l+\frac{1}{2}, l-\frac{1}{2}, l-\frac{3}{2}, l-\frac{5}{2}, \dots $$ To decide where this series terminates, we use the vector inequality $$ |\hat{L}+\hat{S}| \ge \left| |\hat{L}|-|\hat{S}|\right|. $$