TITLE: Sequence converging to different limits
QUESTION [13 upvotes]: In a metric space, is it possible to find a sequence which converges to two different limits with respect to two different metrics?
Obviously the metrics can't be equivalent.

REPLY [27 votes]: With two different metrics? Yes, obviously. (But with two different metrics it is not the same metric space, by definition -- the concept of a metric space includes which metric we're using).
For example take $\mathbb R$ with respectively the standard metric, and the metric
$$ d_2(a,b)=|f(a)-f(b)| \quad\text{where }f(x)=\begin{cases} \pi & \text{if }x=0 \\ 0 & \text{if }x=\pi \\ x & \text{otherwise} \end{cases} $$
Then $a_n=\frac1n$ converges to $0$ in the usual metric but to $\pi$ in the $d_2$ metric.