- #1

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It is said that not every metric comes from a norm.

Consider for example a metric defined on all sequences of real numbers with the metric:

[tex]d(x,y):=\displaystyle\sum_{i=1}^{\infty}\frac{1}{2^i}\frac{|x_i-y_i|}{1+|x_i-y_i|}[/tex]

I can't grasp how can that be.

There is a proof, so could you please give me hints so that I can try and do the proof on my own.

At the moment I can only see that the whole thing is bounded by 1. What does it actually mean when we say "it does not come from a norm"?