The Cauchy-Schwarz Inequality in a Real Inner Product Space

Let $\mathbb{R}$ be the ordered field of real numbers, with the notation of that item, and let $E$ be a real inner product space with inner product $\langle\cdot,\cdot\rangle$ and norm $|\cdot|$, and for a real number $s$ let $|s|$ be its absolute value. Then for all $x,y\in E$,