inner products – Extending sections

I am trying to read Hatcher’s book on vector bundles and I am having trouble understanding a step of the proof of 1.3

We are trying to prove that given a subbundle $E_0 subset E$, there is an orthogonal vector subbundle and in order to do this we consider $m$ local independent sections ${s_i: B to E}_{i in [m]}$ and enlarge this set of sections so that it spans the entire vector space.

I understand how this can be done over one fiber alone, yet I don’t understand how we can guarantee that we can extend this over the entire open set if we do not have more conditions over $B$.

Thank you for any help in advance