Let $K$ be a finite extension of $\mathbb{Q}_p$ and $G$ be a split connected reductive algebraic group over $K$ with Borel $B$. We have the associated Lie algebras $\mathfrak{g}=$Lie$(G)$ and $\mathfrak{b}=$Lie$(B)$.

Let $M$ be a $U(\mathfrak{g})$-module with $N \subset M$ a finite dimensional $K$-module, which is $B$-invariant and generates $M$ as a $U(\mathfrak{g})$-module.

I read that $M$ is then locally $\mathfrak{b}$-finite, i.e $U(\mathfrak{b}) \cdot m \subset M$ is finite dimensional for all $m \in M$, but I have trouble to see this. As $U(\mathfrak{b})$ seems so big for me, I cannot think of a finite basis for $U(\mathfrak{b}) \cdot m \subset M$ by knowing only $N$.