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## Homework Statement

In R^n show that:

[tex]\Vert\overrightarrow{x - y}\Vert \Vert\overrightarrow{x+y}\Vert \leq \Vert\overrightarrow{x}\Vert^{2} + \Vert\overrightarrow{y}\Vert^{2}[/tex]

## Homework Equations

My main attempts have either been from the Triangle Inequality:

[tex] \Vert\overrightarrow{x+y}\Vert \leq \Vert\overrightarrow{x}\Vert + \Vert\overrightarrow{y}\Vert[/tex]

Or from attempts to implement the idea:

sqrt(a^2 + b^2) <= |a| + |b|

## The Attempt at a Solution

Everytime I attempt to do this algebraically, the product on the left becomes a distributive catastrophe and I can't get anything sensible out of it, and attempting to square both sides to get rid of the square root just results in the same catastrophe on the RHS. I tried to represent the left hand side as a dot product but that involved cos(theta) which just complicates the problem. If anyone could point me in the right direction it would be appreciated.