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I am really stuck in the following:

I want to show that for all $\alpha,\beta,\gamma\in\mathbb{R}$ the following is true:

$$|\alpha+\beta-\gamma|+|\alpha+\gamma-\beta|+|\beta+\gamma-\alpha|\ge|\alpha|+|\beta|+|\gamma|$$ I know that I have to proof by cases.

So if I consider $\alpha,\beta,\gamma>0$ don't I have to consider in this case $\alpha+\beta\ge\gamma$ and $\alpha+\beta\le\gamma$, too?

Anybody could help with this inequality? Thanks a lot!

1 Answers 1