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Let's consider a set $X$ with two different metrics (distance function) $d_1, d_2$ on $X$.

Is $\lim_{n\to\infty} d_1(x_n,x)=0 $ equivalent to $\lim_{n\to\infty} d_2(x_n,x)=0$?

I mean, when we can define two different metrics on a set, do the two different metrics give the same limit relation on the set?

  • 3
    Have you tried considering very simple examples, like $X = \mathbb{R}$, $d_1$ the $0-1$ metric and $d_2$ the euclidean metric?2012-09-12
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    The answer is no; the example from @student is a good one.2012-09-12

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