Let $I = ]a,b[ \subset \mathbf{R}$ be a bounded interval, and $L = \partial^n + c_{n-1} \partial^{n-1} + \cdots + c_1 \partial + c_0$, where $c_i \in C^\infty(\overline I)$. Suppose $u \in L^2(I)$ is such that $L u \in L^2(I)$, where $Lu$ is defined in the sense of distributions. Show that $u \in H_n(I)$.
I know how to deal with this when the coefficients are $0$: by induction on $n$, we can show that the solutions of $\partial^n u = 0$ in $L^2(I)$ are just polynomials of order at most $n-1$. Then suppose $u \in L^2(I)$ is such that $\partial^n u = f \in L^2(I)$. Write $$ v(x) = \int_a^x \int_a^{s_1} \cdots \int_a^{s_{n-1}} f(s_n) \; d s_n \ \cdots d s_2 \; d s_1, $$ then $v \in H_n(I)$ and $\partial^n v = \partial^n u$, hence $u = v + p$ for some polynomial $p$.
However, I'm stuck in the general case. I'm aware of various elliptic regularity theorems, however, this question is from an introductory text on distribution theory and is asked before such theorems are discussed. So I think there should be a more elementary proof...