Trouble understand proof of spectral theorem

Question

I'm reading through this proof of the real spectral theorem. I don't understand the last line of "lucky fact 2" - why must $\overrightarrow{u}$ have been listed in the $v_{i}$?

Answer 1

The point is that you get a contradiction: there turns out to be an eigenvector for eigenvalue $\lambda$, not in the span of the $v_1,\dots, v_r$, if $C\vec u=\lambda \vec u$. So the eigenspace for $\lambda $ has $1$ greater dimension than supposed.