Chapter 6.7, Problem 20E

Let $u=\left[\begin{array}{c}a\\ b\end{array}\right]$ and $v=\left[\begin{array}{c}1\\ 1\end{array}\right]$. Use the Cauchy-Schwarz inequality to show that

${\left(\frac{a+b}{2}\right)}^2\le \frac{a^2+b^2}{2}$