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While working with some inequalities I've noticed that combining the Cauchy-Schwartz inequality with some others we get the following:

$|x|+|y|\ge|x+y|\ge |x-y|\ge||x|-|y||\ge|x|-|y|$

Does this inequality hold, or is there something I'm missing here?

3 Answers 3

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$|x| + |y| \ge |x| - |y|$ holds for all real numbers, but your derivation of that inequality is invalid: $$ |x+y| \ge |x - y| $$ does not hold if $x$ and $y$ have opposite signs (e.g. $x=1$, $y=-1$).

As you figured out in the meantime, removing $|x+y|$ or $|x-y|$ from the inequality chain makes it correct: $$ |x|+|y|\ge |x+y|\ge||x|-|y||\ge|x|-|y| \\ |x|+|y|\ge |x-y|\ge||x|-|y||\ge|x|-|y| $$ which the same inequalities via the substitution $y \to -y$.

(Of course you don't need those triangle inequalities in order to prove $|x| + |y| \ge |x| - |y|$, as @Ant demonstrated.)

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    Yes, this makes sense, any ideas on how the inequality should be? Maybe it should be in two parts? – 2017-02-16
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    @Michael: What inequality? What are you actually trying to prove? There is no relationship between $|x+y|$ and $|x-y|$ which holds for all real numbers. – 2017-02-16
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    So to sort things out here it goes somethin like this right?: $|x|+|y|\ge|x+y|\ge||x|-|y||\ge|x|-|y|$ $|x|+|y|\ge |x-y|\ge||x|-|y||\ge|x|-|y|$ – 2017-02-16
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    @Michael: Yes, those inequality chains are both correct. – 2017-02-16
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Your inequality is equivalent to $$|y| \ge -|y|$$, that is $$|y| \ge 0$$

So yes, it holds

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    Hmm, I'm not quite sure how it is equivalent to that – 2017-02-16
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    @Michael Do you agree that $|y|\ge-|y|$? Then add $jx|$ to both sides. – 2017-02-16
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To show that $|x|+|y|\ge|x+y|\ge |x-y|\ge||x|-|y||\ge|x|-|y|$ :

  1. $|x|+|y|\ge|x+y|$ by the triangle inequality
  1. $|x+y|\ge |x-y|$ is false.
  1. $|x-y|\ge||x|-|y||$ by reverse triangle inequality
  1. $||x|-|y||\ge|x|-|y|$ is of the form $|x| \ge x$, which is clearly true
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    (2) is wrong for $x=1, y=-1$. – 2017-02-16
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    Your revised version is not correct either: $x=2, y=-3$. – 2017-02-16
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    @MartinR *sigh*. Yep. – 2017-02-16