I am reading an article and it states that if $F$ is $\mathbb{R}$ or $\mathbb{C}$, $X$ is a Compact space and $\xi$ a (finite dimensional and locally trivial) $F$-vector bundle, then there exists another $F$-vector bundle $\xi'$ (fin. dim and loc. trivial) such that $\xi \oplus \xi’$ is isomorphic to a trivial vector bundle.
Any ideas on how to prove this?