Is this proof correct? Establishing that $|x+y| \geq | |x| - |y| |$

Asked Dec 18 '18 at 23:26

Active Dec 18 '18 at 23:26

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Using that $|a|+|b|\geq|a+b|$ \begin{align} |-x|+|x+y| \geq |-x+x+y| = |y|\\ |-y|+|x+y| \geq |-y+x+y| = |x| \end{align}

Substracting $|-x|$ from the first inequality and $|-y|$ from the second:

\begin{align} |x+y| \geq |y|-|-x| \\ |x+y| \geq |x| - |-y| \end{align}

Using the fact that if $a \geq b$ and $a \geq -b$ then $a \geq |b|$.

\begin{align} |x+y| \geq ||x| - |-y|| = ||x| - |y|| \end{align}

Is the above correct?

asked Dec 18 '18 at 23:26

Pinteco

3

The proof is good! – Theo Bendit Dec 18 '18 at 23:34
1

Seems good enough! – Rebellos Dec 18 '18 at 23:40
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Nice work. See also Help checking proof of $|x| - |y| \leq |x+y|$ – amWhy Dec 18 '18 at 23:57
Perfect. But I would have typed $|(|x|-|y|)|$ or $|; |x|-|y|;|$(etc.) to make it easier to read, and to avoid confusion with the functional-analysis symbol $||z||$ (also written $|z|,$ coded as |z|). You can use ; and , to add a little space between key-strokes. – DanielWainfleet Dec 19 '18 at 03:16

0 Answers0