- #1

swevener

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## Homework Statement

Prove the Schwarz inequality by first proving that

[tex](x_{1}^{2} + x_{2}^{2})(y_{1}^{2} + y_{2}^{2}) = (x_{1} y_{1} + x_{2} y_{2})^{2} + (x_{1} y_{2} - x_{2} y_{1})^{2}.[/tex]

## Homework Equations

[tex]x_{1} y_{1} + x_{2} y_{2} \leq \sqrt{x_{1}^{2} + x_{2}^{2}} \sqrt{y_{1}^{2} + y_{2}^{2}}[/tex]

## The Attempt at a Solution

I'm not sure if my logic is right. I did the little proof above, and with that I can say

[tex]-(x_{1} y_{2} - x_{2} y_{1})^{2} \leq (x_{1} y_{1} + x_{2} y_{2})^{2}.[/tex]

Can I then sweep the LHS under the zero and say

[tex](x_{1} y_{1} + x_{2} y_{2})^{2} \leq (x_{1}^{2} + x_{2}^{2})(y_{1}^{2} + y_{2}^{2}),[/tex]

then take the square root to finish the proof?