MHB A proof of a general property of norms

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The discussion revolves around proving the inequality | ||x|| - ||y|| | <= ||x - y|| for any norm on any vector space. The original poster expresses confusion about how to approach the proof, feeling uncertain about the fundamental properties of norms that could apply. Respondents suggest that the triangle inequality is key to the proof, specifically by considering the expression ||x - y + y||. This approach helps clarify how to manipulate the norms involved to establish the desired result. Overall, the conversation emphasizes the importance of the triangle inequality in proving the general property of norms.
gucci1
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So I have been asked to prove a result that is supposedly valid for any norm on any vector space. The statement to prove is: | ||x|| - ||y|| | <= ||x - y||

The problem is, I have no idea where to start with this proof. Maybe I'm missing some fundamental property of norms, but it seems that having to prove this for any norm on any vector space rules out any definitions that I might try to apply :-/

Sorry for not having anything to go on here, but I'm lost. Thank you all very much for any help you might be able to offer!
 
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This follows from the triangle inequality. Consider the quantity $\| x-y+y \|$.
 

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