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For x, y ∈ R, prove that ||x| − |y|| ≤ |x − y|, (Hint: consider x = y + (x − y)).​

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Using the triangular inequality (and the given hint!), we have


|x|=|(x-y)+y|\leq|x-y|+|y|\implies|x|-|y|\leq|x-y|

Similarly,


|y|=|(y-x)+x|\leq|x-y|+|x|\implies|y|-|x|\leq|x-y|\implies -|x-y| \leq |x|-|y|

We managed to bound the quantities in this fashion:


-b\leq a\leq b \implies |a|<b

And thus we have


||x|-|y||\leq |x-y|

User Yeshyyy
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