Exercise 37, § 1

Let $X$ be a differential manifold of class $C^r$ where $1\leqslant r\leqslant \infty$, paracompact, locally of finite dimension, and let $E$ be a vector bundle over $X$ of class $C^r$, locally of finite dimension. Prove that the space $\mathscr{S}^r(X; E)$ possesses the approximation property.