Integer spectrum of the number operator
= Integer spectrum of the number operator
If $N|\nu\rangle=\nu|\nu\rangle$, then $A$ lowers the eigenvalue by one and $\|A|\nu\rangle\|^2=\nu\|\nu\rangle\|^2$. Positivity forces the lowering ladder to terminate at eigenvalue zero, so every eigenvalue is a nonnegative integer.