Vector bundle over compact implies summand of trivial

Statement

Verbal statement

Any real vector bundle over a compact Hausdorff space is a direct summand of a trivial vector bundle.

Symbolic statement

If $p:E\to B$ is a real vector bundle with $B$ a compact Hausdorff space, then there exists a vector bundle $E'$ such that $E\oplus E'$ is a trivial vector bundle over $B$.