Can I find a finite increasing filtration for every $V\in\mathfrak{O}$?

Question

Let $V\in{O}$, I want to proof there exists a finite increasing filtration by submodules $0=V_0\subset V_1\subset\cdots\subset V_n=V$ such that $V_{i+1}/V_i$ is a highest weight module. Since $V\in {O}$ than there exists a finite number of elements $\lambda_1\cdots\lambda_s$ such that $$P(V)\subset\bigcup_{i=1}^s D(\lambda_i)$$ so if I set $V_1=U(\mathfrak{g}(A))v$, $v\in V_{\lambda_1}$ and than how can I do?