# 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.