<s>[INST] Michael Spivak's Calculus - Chapter 1, Problem 19 Problem 19. The fact that ${a^2}\ge{0}$ for all the numbers a, elementary as it may seem, is nevertheless the fundamental idea upon which most important inequalities are ultimately based. The great-grandaddy of all inequalities is the Schwarz inequality:
${x_{1}y_{1}+x_{2}y_{2}}\le{\sqrt{x_{1}^2+x_{2}^2}\sqrt{y_{1}^2+y_{2}^2}}$
The three proofs of the Schwarz inequality outlineed below have only one thing in common - their reliance on the fact that $a^2\ge{0}$ for all $a$.
(a) Using Problem 18, complete the proof of the Schwarz inequality.
Solution.
This is the proof I've written.
\begin{array}{ll}
(x_{1}y_{2}-x_{2}y_{1})^{2} & \ge0\\
x_{1}^{2}y_{2}^{2}-2x_{1}x_{2}y_{1}y_{2}+x_{2}^{2}y_{1}^{2} & \ge0\\
x_{1}^{2}y_{2}^{2}+x_{2}^{2}y_{1}^{2} & \ge2x_{1}x_{2}y_{1}y_{2}\\
2x_{1}x_{2}y_{1}y_{2} & \le x_{1}^{2}y_{2}^{2}+x_{2}^{2}y_{1}^{2}\\
x_{1}^{2}y_{1}^{2}+x_{2}^{2}y_{2}^{2}+2x_{1}x_{2}y_{1}y_{2} & \le x_{1}^{2}y_{1}^{2}+x_{2}^{2}y_{2}^{2}+x_{1}^{2}y_{2}^{2}+x_{2}^{2}y_{1}^{2}\\
(x_{1}y_{1}+x_{2}y_{2})^{2} & \le(x_{1}^{2}+x_{2}^{2})(y_{1}^{2}+y_{2}^{2})\\
x_{1}y_{1}+x_{2}y_{2} & \le\sqrt{(x_{1}^{2}+x_{2}^{2})}\sqrt{(y_{1}^{2}+y_{2}^{2})}
\end{array}
(b)Prove the Schwarz inequality by using $2xy\le{x^2+y^2}$ with
$x=\displaystyle{\frac{x_{i}}{\sqrt{x_{1}^2+x_{2}^2}}}$, $y=\displaystyle{\frac{y_{i}}{\sqrt{x_{1}^2+x_{2}^2}}}$
first for $i=1$ and then for $i=2$. 
I am not able use the inequality to come up with another proof. Could someone tell me the way, how to approach this. 
Also, what are some of the applications of Cauchy-Schwarz inequality? I did google it, but would love to hear it from a mathematician!
 [/INST] We have
\begin{align}2x_iy_i &\le \frac{x_i^2 \sqrt{y_1^2 + y_2^2}}{\sqrt{x_1^2 + x_2^2}} + \frac{y_i^2 \sqrt{x_1^2 + x_2^2}}{\sqrt{y_1^2 + y_2^2}}\\
&\le \frac{x_i^2 (y_1^2 + y_2^2) + y_i^2 (x_1^2 + x_2^2)}{\sqrt{x_1^2 + x_2^2}\sqrt{y_1^2 + y_2^2}}\\
&\le \frac{x_i^2 (y_1^2 + y_2^2) + y_i^2 (x_1^2 + x_2^2)}{\sqrt{x_1^2 + x_2^2}\sqrt{y_1^2 + y_2^2}}\\
\to 2 \sum_{i\in\{1,2\}} x_iy_i &\le \frac{(x_1^2 + x_2^2) (y_1^2 + y_2^2) + (y_1^2 + y_2^2) (x_1^2 + x_2^2)}{\sqrt{x_1^2 + x_2^2}\sqrt{y_1^2 + y_2^2}}\\
\to \sum_{i\in\{1,2\}} x_iy_i &\le \frac{(x_1^2 + x_2^2) (y_1^2 + y_2^2)}{\sqrt{x_1^2 + x_2^2}\sqrt{y_1^2 + y_2^2}}\\
\to \sum_{i\in\{1,2\}} x_iy_i &\le \sqrt{x_1^2 + x_2^2}\sqrt{y_1^2 + y_2^2}\end{align}
QED
</s>