Suppose I have a set with two metrics, which induce distinct topologies, (so neither is contained in the other). There should exist a sequence which converges in both topologies, but to different points (otherwise all sequences converge to the same point in both metrics, which then implies the topologies are equal). I'm having trouble coming up with such an example.
Can anyone kindly provide a hint, or is this a bad question?
Thanks!
It is a "bad question", in that the assumption that neither metric topology is contained in the other does not imply that there is a sequence converging in both topologies, but to different points. Example: on the set $\mathbb{Z} \cup \{+\infty\}\cup\{-\infty\}$ consider the metric topology $\tau_1$ where all points are open but $+\infty$ , and $+\infty$ is the limit of the sequence $x_n:=n$; also, the metric topology $\tau_2$ where all points are open but $-\infty$ , and $-\infty$ is the limit of the sequence $x_n:=-n$ . Here, any sequence that converges in both topologies is eventually constant, so the limit is the same, and there are sequences converging in either topology and non-converging in the other.
Take $\mathbb{Q}$ and consider two different norms on it: one is simply Euclidean and another is $p-$ adic. Take the sequence $x_n=\frac{p^n}{p^n-1}.$ Clearly $x_n\to 1$ in Euclidean norm and $x_n\to 0$ in $p-$ adic.
