Sections of tensor bundle are tensor product of sections

Question

Given $E,F$ vector bundles over a manifold $M$, I would like to know a proof of $\Gamma(E\otimes F) = \Gamma(E) \otimes_{C^\infty(M)} \Gamma(F)$. Where $\Gamma$ denotes the smooth sections over $M$. Thanks

Answer 1

The statement is trivially true if $E$ is a trivial vector bundle. Note also that if the statement is true for $E$, then it is also true for any direct summand of $E$. The conclusion follows from the fact that any vector bundle on a manifold is a subbundle of a trivial vector bundle.