- #1

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## Homework Statement

Show that [tex]\forall a,b \in R[/tex]:

[tex]\left|ab\right|\leq\frac{1}{2}(a^{2}+b^{2})[/tex]

## Homework Equations

Triangle Inequality seems to be useless.

## The Attempt at a Solution

[tex](a+b)^{2}=a^{2}+b^{2}+2ab[/tex]

[tex]2ab=(a+b)^{2}-(a^{2}+b^{2})[/tex]

[tex]ab=\frac{1}{2}(a+b)^{2}-\frac{1}{2}(a^{2}+b^{2})[/tex]

[tex]\left|ab\right|=\left|\frac{1}{2}(a+b)^{2}-\frac{1}{2}(a^{2}+b^{2})\right|[/tex]

[tex]\left|ab\right|=\left|\frac{1}{2}(a^{2}+b^{2})-\frac{1}{2}(a+b)^{2}\right|[/tex]