I am trying to understand if it is true that: $$|x|-|y|\leq|x-y|\leq|x|+|y|$$ I know that this is a topic maybe discussed already many times...maybe I am not so good on searching for question already done..so in case this is a question already done can you tell me if the inequality holds and eventually can you give me the link of the that question?
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https://en.wikipedia.org/wiki/Triangle_inequality – Albus Dumbledore May 16 '21 at 09:34
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Yes, it's correct. In fact it can be extended to complex numbers. It is a consequence of the triangle inequality. – Ritam_Dasgupta May 16 '21 at 09:35
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@Ritam_Dasgupta thanks! In the question suggested seems that on the left I have to consider the absolute value of the difference...it is right also without it as indicated in my question? – lay May 16 '21 at 09:43
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You can use that $|x|-|y| \le | |x|-|y| |$ to get from the version with abs val to the version without. – coffeemath May 16 '21 at 09:54
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2Or simply observe that $|x| = |x - y + y| \leq |x - y| + |y|$ and subtract $|y|$ from both sides. (Essentially proving just the half of the absolute value version that you need.) – May 16 '21 at 10:20