Every smooth vector bundle admits a smooth bundle metric

Index an open cover by all pairs $(U_{\alpha },{\tau }_{\alpha })$ supplied by the bundle's local trivializations, so each cover member comes with its trivialization and no separate family of frame choices is needed. Pull back the Euclidean inner product on ${\mathbb{R}}^r$ through ${\tau }_{\alpha }$ to obtain a smooth local bundle metric $h_{\alpha }$ on $E{\mathrm{\mid }}_{U_{\alpha }}$.

Under the stated Countable Choice, [L1] gives a smooth partition of unity $({\rho }_{\alpha })$ indexed by these same trivialization pairs, with $\mathrm{supp}{\rho }_{\alpha }\subseteq U_{\alpha }$. Define ${\rho }_{\alpha }{h}_{\alpha }$ on $U_{\alpha }$ and extend it by zero outside $U_{\alpha }$; this is smooth because every point outside the closed support of ${\rho }_{\alpha }$ has a neighbourhood where the term vanishes. The sum $h:={\sum }_{\alpha }{\rho }_{\alpha }{h}_{\alpha }$ is locally finite and smooth. At each point $p$, some ${\rho }_{\alpha }(p)>0$, all weights are nonnegative, and ${\sum }_{\alpha }{\rho }_{\alpha }(p)=1$, so $h_p$ is positive definite. Thus $h$ is a smooth bundle metric.