Suppose the eigenstate of vec$(L{)}^2$ and $L_z$ operators is $|l,m$:$|$ satisfying vec$(L{)}^2|l,m$:$|$ and $L_z|l,m$:$|,$ prove that $L_{-}|l,m$:$|,$ where $L_{-}=L_x-i{L}_y.$

Following $47.,$ explain why $l$ should be integers or half integers.