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Every object $M$ of Category O has a finite composition series of a module. Choose weight-vector generators and let $T$ be the sum of their finite-dimensional weight spaces. The centre preserves $T$; the kernel of its action on $T$ has finite codimension and, by centrality, annihilates all generators and hence all of $M$. Thus only finitely many central characters can appear in subquotients. The Harish-Chandra isomorphism makes each such character correspond to one finite Weyl orbit of shifted highest weights. Let $S$ be the resulting finite set of possible highest weights.

Every nonzero subquotient has a highest-weight vector from its bounded-above support, and a common central eigenvector can be chosen in the finite-dimensional highest-weight space. Its weight belongs to $S$. The integer $d(N)={\sum }_{\nu \in S}\dim N_{\nu }$ is therefore finite, additive on short exact sequences and positive for every nonzero subquotient $N$. Each strict refinement of a submodule chain consumes at least one unit of $d(M)$, bounding its length and proving the claim.

The centre commutes with $\mathfrak{h}$, so every nonzero representation weight component of the common eigenvector $m$ has the same central character $\theta$. Pick such a component $m_{\lambda }$. In Category O, its $U({\mathfrak{n}}_{+})$-span is finite-dimensional and has finitely many weights. Choose a maximal representation weight in that span and a nonzero vector $v$ of that representation weight, say $\nu$. Every positive root operator kills $v$, since it would raise the representation weight within the same finite span. Centrality ensures $zv=\theta (z)v$.

The universal property supplies a nonzero map $M^{\nu +\rho }\to M$ taking its highest vector to $v$. The centre acts on that image both by $\theta$ and by ${\theta }_{\nu +\rho }$. Hence

Use the standard defining properties of Category O: finite generation over $U(\mathfrak{g})$, a direct sum of finite-dimensional weight spaces, and locally finite positive root action. In particular the weights lie in a finite union of downward root cones. Submodules and quotients retain the representation weight bounds and finite weight spaces; finite generation of submodules follows from the Poincaré-Birkhoff-Witt theorem and the noetherian property of $U(\mathfrak{g})$.

Choose finitely many weight vector generators. The sum $T$ of their full weight spaces is finite-dimensional and preserved by the centre. The kernel of its central action has finite codimension in $Z(U(\mathfrak{g}))$ and kills the generators, hence, by centrality, all of $M$. Thus only finitely many central characters ${\theta }_1,\dots ,{\theta }_r$ can occur in any subquotient: they are the characters of a fixed finite-dimensional commutative quotient algebra of the centre.

By the Harish-Chandra isomorphism each ${\theta }_i$ determines one finite Weyl orbit of shifted weights. Let $S$ be the finite union of these orbits after subtracting $\rho$. Every nonzero subquotient has a nonzero highest-weight vector, using the bounded-above weights. Within its finite-dimensional highest weight space, the commuting central operators have a common eigenvector. Part (e) and the central-character restriction show that its highest weight belongs to $S$.

Define for any subquotient $N$It is finite, additive in short exact sequences, and positive whenever $N\ne 0$. Hence each strict step of a submodule chain consumes at least one unit of $d(M)$. Refining a nonsimple factor into a proper submodule and quotient cannot continue more than $d(M)$ times. It ends with simple factors, provingThe argument proves the assertion for every object of $\mathcal{O}$, rather than just for a cyclic highest weight object.