What is a smooth vector bundle over a manifold $M$ ?

Assuming the existence of "bump functions", prove that every compact manifold embeds in some Euclidean space $\mathbb{R}^{n}$.

By choosing an inner product on $\mathbb{R}^{n}$, or otherwise, deduce that for any compact manifold $M$ there exists some vector bundle $\eta \rightarrow M$ such that the direct sum $T M \oplus \eta$ is isomorphic to a trivial vector bundle.