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I am wondering whether the concept of two metrics defining the "same convergent sequences" (for instance, when we are interested in the metrics being topologically equivalent) also includes the respective limits of the sequences.

Suppose we have a set $X$ and two metrics $d_1,d_2$ defined on it. Furthermore, assume for all sequences $(x_n)$ in $X$ that $d_1(x_n,x)\rightarrow 0$ if and only if $d_2(x_n,x^\prime)\rightarrow 0$, i.e., a sequence that converges to some limit $x$ in $d_1$ also converges to some limit $x^\prime$ in $d_2$. Does this imply $x=x^\prime$, i.e., the sequences having the same limit?

Thus, both metrics define the same sequences in $X$ as being convergent sequences, but does that also include that they have the same limit? If not, is there a counterexample?

mltm97
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    If $x_1,x_2,x_3,\ldots \rightarrow x$, then $x_1,x,x_2,x,x_3,x,\ldots \rightarrow x$ also. But the latter sequence can't converge to anything besides $x$. – mjqxxxx Nov 04 '22 at 15:42
  • I think this is answered here: https://math.stackexchange.com/q/368334/42969 – Martin R Nov 04 '22 at 15:52

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