Eventually constant sequence under discrete metric but not under norm

I've been trying to come up with an example of a sequence which is eventually constant under the discrete metric, but has terms come closer together under, say, the Euclidean norm. Do such sequences exist? I have a problem coming up with one.

• "Eventually constant"-ness of a sequence is independent of metric or norm
– edm
Oct 3 '17 at 5:15
• Perhaps you meant to say divergent instead of eventually constant?
– wjm
Oct 3 '17 at 5:22

A sequence $(x_n)$ is eventually constant if there exists $N$ such that $x_n=x_m$ for all $m,n\geq N$. This definition does not involve any metric, and so whether a sequence is eventually constant does not depend on what metric you use.

• Does this imply that if a sequence is convergent to the same point under two different metrics, then these metrics are necessarily equivalent? Oct 3 '17 at 5:34
• No...why would it imply that? Oct 3 '17 at 6:15
• Here (math.stackexchange.com/questions/2455092/…) I was given a hint, but I guess it was quite wrong. Or I misinterpreted it somehow. Oct 3 '17 at 6:19