I wonder why the total angular momentum $$J=J_1 +J_2 $$ is given in the range of $$ |j_1 - j_2 | \leq j \leq j_1 +j_2 $$ Of course we can verify this in the course of finding Clebsh-Gordan Coefficients. If we start from 'top state' of $j=j_1 +j_2 $ and follow the step of lowering and orthonormalizing and so on, we can see that there is no more possible total angular momentum for $j$ when we reach $j= |j_1 -j_2|$.
But this is not a 'explanation' for the very reason why $ |j_1 - j_2 | \leq j \leq j_1 +j_2 $ should be holds.. It seems that it has to do with triangle inequality but eigenvalues of $J^2$ are $\hbar^2 j(j+1) $ and those of $J_1 $ and $J_2$ are $\hbar^2 j_1 (j_1 +1) $ and $\hbar^2 j_2 (J_2 +1)$ respectively so... what to do?