Any smooth complex line bundle over a smooth manifold is a subbundle of a trivial bundle

Question

In the book Vector Bundles and K-Theory of Hatcher, it is proven (Proposition 1.4) that for any (continuous) vector bundle $E\to B$ with $B$ a compact Hausdorff space, $E$ is a subbundle of a trivial bundle. Also it is remarked that this can fail when $B$ is noncompact: the canonical line bundle over $\Bbb RP^\infty$ is an example that fails.

But I am curious about the following special situation: Suppose $L$ is a (smooth) complex line bundle over a smooth manifold $M$. Then is it true that $L$ is a subbundle of a trivial (complex) bundle over $M$?

Answer 1

Yes, provided $M$ is finite dimensional and smooth. The key property is that a vector bundle over such a manifold has can be covered by a finite set of local trivializations $\{(U_i,\varphi_i)\}_{i=1}^N$ (see this answer for details). Armed with these trivializations, one can embed $L$ in $M\times\mathbb{C}^N$ using partitions of unity.