For a while I've been reading J.E.Humphreys's book "Representations of semisimple Lie algebras in the BGG category O" under the impression that any module in O has a finite generating set composed of highest weight vectors. Now I've realised that I'm lacking a proof. So what would serve as a counterexample? Or has my assumption been correct for some reason?

For a while I've been reading J.E.Humphreys's book "Representations of semisimple Lie algebras in the BGG category $\mathcal O$" under the impression that any module in $\mathcal O$ has a finite generating set composed of highest weight vectors. Now I've realised that I'm lacking a proof. So what would serve as a counterexample? Or has my assumption been correct for some reason?

I apologize if the answer can be found further on in the book itself!

For a while I've been reading J.E.Humphreys's book "Representations of semisimple Lie algebras in the BGG category O" under the impression that any module in O has a finite generating set composed of highest weight vectors. Now I've understood that I'm lacking a proof. So what would serve as a counterexample? Or has my assumption been correct for some reason?

For a while I've been reading J.E.Humphreys's book "Representations of semisimple Lie algebras in the BGG category O" under the impression that any module in O has a finite generating set composed of highest weight vectors. Now I've realised that I'm lacking a proof. So what would serve as a counterexample? Or has my assumption been correct for some reason?