This proof comes from Linear Algebra by Friedberg, Spence, Insel. I don't understand the last step in this proof, which I highlight with a purple line. I understand that $\sum_{j=1}^n a_j^2 H(v_j, v_j) = \sum_{j=p+1}^r H(v_j, v_j) $ since $a_i = 0$ for $i \le p$ and $H(v_i, v_i) = 0 $ for $i>r$. But, why $b_i =0$ for $i=1, ... , p$?
I appreciate if you give some help, and I apologize to ask you for reading a long proof.
The last term should not be $\sum_{j=p+1}^{r} b_j^2 H(w_j,w_j)$ but rather it should be
$\sum_{j=1}^{q} b_j^2 H(w_j,w_j) + \sum_{j=r+1}^{n} b_j^2 H(w_j,w_j)$, which is non-negative.