The question tells me that i can use $x=(x+y)-y$ and $|x+y|≤|x|+|y|$ to prove. But i don't know where to use $x=(x+y)-y$.
Till now,i only get that $-|x|-|y|≤|x+y|$ from the use of $|x|≤A$,$-A≤x≤A$
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$$ \lvert x \rvert = \lvert (x+y)-y \rvert \\ \leq \lvert x+y \rvert + \lvert -y \rvert = \lvert x+y \rvert + \lvert y \rvert,. $$ the first equality being the first part of the hint, the second being the inequality in the hint, albeit with symbols with different names (write $z=x+y$, $w=-y$, then one has $\lvert z+w \rvert \leq \lvert z \rvert + \lvert w \rvert$ using the hint: it doesn't just mean the original $x$ and $y$, but any numbers).
Chappers
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Good to see that you're still alive. (+1) for the answer. Remember to good ole days when the site had a plethora of cool analysis questions (neat series, products, integrals, and beyond)? What the heck happened? – Mark Viola Sep 24 '17 at 23:44
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Thanks.I have sloved my problem. – will01805 Sep 24 '17 at 23:45
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@MarkViola IK,R? Been quite a disappointing last few months on here. Let's hope things improve once the academic year gets properly going. – Chappers Sep 25 '17 at 00:23
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@Chappers Just curious ... What did you mean by "IK, R?" – Mark Viola Sep 25 '17 at 04:35
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@MarkViola "I know, right?" – Chappers Sep 25 '17 at 11:14
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Ah, ISN. How did I miss that? – Mark Viola Sep 25 '17 at 13:02