Vector bundle over compact implies summand of trivial

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 $$B$$.