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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?

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    The answer is no; the example from @student is a good one.2012-09-12

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If two metrics are equivalent, i.e., if there are two constants $C_1$ and $C_2$ such that $C_1 d_1(x,y) \leq d_2(x,y) \leq C_2 d_1(x,y),$ then convergence in the two metrics is the same.

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    This doesn't answer the question asked...2012-09-12
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HINT: Let $X$ be any set. Define

$d:X\times X\to\Bbb R:\langle x,y\rangle\mapsto\begin{cases}0,&\text{if }x=y\\1,&\text{if }x\ne y\;.\end{cases}$

Show that this is a metric on $X$. (It’s the $0$-$1$ metric mentioned by student in the comments.) What sequences converge in this metric? (They’re pretty boring sequences.) Can you find an $X$ that has another topology with different convergent sequences?