I am reading the following proof of Cauchy Schwarz inequality.
Statement: Let $V$ be an inner product space and $u, v \in V$. Then $|\langle u, v\rangle |\le \|u\| \|v\|$.
Proof of the above inequality goes as follows:
Proof: If $u$ or $v$ is zero then equality holds. So assuming that $u$ and $v$ are non zero.
Now consider the inner product $\langle v - \lambda u, v -\lambda u\rangle ~\geq 0$, where $\lambda = \frac{\langle v, u\rangle}{\langle u, u\rangle} = \frac{\overline{\langle u, v\rangle }}{\|u\|^2}$. And after that an easy proof is given. I have also attached the complete proof of this inequality. 
I am confused why this particular value of $\lambda$ is chosen? Why $\lambda$ is not an arbitrary scalar?
Thank you in advance.