Problem 2-C (Existence theorem for Euclidean metrics)

Proof. Let $\{U_{\alpha }\}$ be an open cover of the base space $M$ such that the vector bundle $(E,M,\pi )$ is trivial over each $U_{\alpha }$. Over each $U_{\alpha }$, we can equip ${\pi }^{-1}(U_{\alpha })\simeq U_{\alpha }\times R^{\mathrm{\text{rank}}(E)}$ with a Euclidean metric ${\text{⟨⋅,⋅⟩}}_{\alpha }$. By paracompactness, there is a partition of unity $\{{\psi }_{\alpha }\}$ subordinate to $\{U_{\alpha }\}$, so $⟨\cdot ,\cdot ⟩:=\sum <!--\; FUNCTION\; APPLICATION\; -->{\psi }_{\alpha }{\text{⟨⋅,⋅⟩}}_{\alpha }$ is a Euclidean metric on $E$. □