Vector bundle over paracompact base admits inner product

Statement
Let $$E$$ be a real vector bundle over a paracompact Hausdorff space $$B$$. Then, we can define an inner product (a positive-definite symmetric bilinear form) on each of the fibers of $$E$$, that varies continuously in the basepoint.

Proof
We do the following steps:


 * Choose a system of local trivializations for the bundle $$E$$. This gives an open cover of $$B$$. Select an inner product on each of the local trivializations.
 * Using the fact that $$B$$ is paracompact Hausdorff, construct a partition of unity subordinate to this open cover.
 * Now for any point, take the sum of the inner products arising from each of the local trivializations at the point, weighted by the corresponding function in the partition of unity