I was given the hint to split it into two cases ($|a| - |b|$ being positive and negative) and then use the triangle inequality. However, since the triangle inequality says that $|a+b|$ is less than or equal to $|a| + |b|$, so I don't see how I can use that to help since I'm dealing with subtraction. I've tried using the definition of absolute value, and I was able to find that $||a| - |b||$ = $|a| - |b|$ in my first case, but I couldn't do much with that.
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1You may change the subtraction $x-y$ into the addition $x+(-y)$ (this is the definition of subtraction, after all), which should enable the triangle inequality. – Arthur Jan 24 '16 at 09:16
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4See also this question and other posts linked there. – Martin Sleziak Aug 01 '16 at 05:18
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You may write $$ x=x-y+y $$ giving, by the triangle inequality, $$ |x|\leq |x-y|+|y| $$ or
$$ |x|-|y|\leq|x-y| $$
then do the same starting this time with $$y=y-x+x.$$
Olivier Oloa
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Instead of your last step we can do this : after getting $|x|-|y| \le |x-y|$, we can just swap $x$ & $y$ in it to get $|y|-|x| \le |y-x|=|x-y|$ – Error 404 Jan 24 '16 at 09:27
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@VikrantDesai Right, but "swapping" $x$ and $y$ is fact doing the same thing as starting with $y$. – Olivier Oloa Jan 24 '16 at 09:30
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You may prove $(|a|-|b|)^2 \leq (a-b)^2$, since $2|a||b| \geq 2ab$. By taking positive absolute values, you obtain $||a|-|b|| \leq |a-b|$.
bing
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