We prove Broyden's theorem by induction on the dimension. We assume the result holds up to rank $n - 1$ and we write $O$ in the form of a block matrix
$$O = \left( \begin{array} { l l }
P & r \\
{ } ^ { t } q & \alpha
\end{array} \right)$$
where $P \in M _ { n - 1 } ( \mathbb { R } )$ and thus $r , q \in \mathbb { R } ^ { n - 1 }$ and $\alpha \in \mathbb { R }$.
Treat the case $| \alpha | = 1$.