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I was trying to prove that every normed space is a metric space, and the first three proprierties came natural. However, when faced with proving the triangle inequality I had a bit of problems. I tried playing with the properties but got nowhere. Now I'm beginning to have doubts over the fact that it is true at all. Is it? If so, could you provide a proof (which, by the way, I haven't found googling)?

Thank you in advance.

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Norms satisfy $$\lVert x+y\rVert \leqslant \lVert x\rVert +\lVert y\rVert$$

Then let $x=x'-y',y=y'-z'$. What we're saying is every norm induces a metric. But note the converse is not true.

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  • $\begingroup$ Oh, thank you. Yes, I had noticed that. Sorry for not being able to upvote. $\endgroup$ – Gennaro Marco Devincenzis Sep 24 '13 at 22:07

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