Problem
Problem 2.17 Let $(E, \pi, M)$ be a $C^{\infty}$ vector bundle with fibre $\mathbb{F}^{n}$, where $\mathbb{F}=$ $\mathbb{R}, \mathbb{C}$ or $\mathbb{H}$. Prove that the homotheties
$$
h: \mathbb{F} \times E \rightarrow E, \quad(\lambda, y) \mapsto h(\lambda, y)=\lambda y
$$
are $C^{\infty}$.