I am reading R.O.Wells' "Differential Analysis on Complex Manifolds" to learn the theory of elliptic operators on compact manifolds and have a problem understanding the proof of Theorem 4.9 on page 139 (Elliptic Regularity).
In order to try understand the proof better, I am trying to formulate a special case of it purely in terms of Cauchy sequences of smooth functions. Here is my formulation: Suppose $L\in\text{Diff}_k(E,F)$ is an elliptic operator between sections of vector bundles $E,F$ on a compact manifold $X$. If $\xi_n\in\Gamma(X,E)$ are such that $\xi_n$ are Cauchy with respect to the $W^s(E)$-norm, and $L\xi_n\to 0$ in the $W^{s-k}(F)$-norm, then in fact $\xi_n$ is Cauchy in the $C^r$ norm for every $r\ge 0$ and so converges to a smooth $\xi\in\Gamma(X,E)$ in the $C^\infty$ topology with $L\xi = 0$.
Firstly, is my formulation correct? If yes, am I right in interpreting the proof as an argument which shows that $\xi_n$ being Cauchy in the $W^s(E)$-norm implies that it is Cauchy in the $W^{s+1}(E)$-norm and then using induction and the Sobolev inequalities to conclude $C^r$ convergence? If this interpretation is right, then I am stuck:
Let $P$ be a pseudodifferential operator such that $P\circ L-I_E$ and $L\circ P-I_F$ are of order $-1$. Then, note that $PL\xi_n-\xi_n$ is Cauchy in the $W^{s+1}(E)$-norm. How do I conclude from this that $\xi_n$ are in fact Cauchy in the $W^{s+1}(E)$-norm?