# Are there any non constant - convergent sequence with discrete metric?

The sequence $\frac{1}{n}$ is convergent under euclidean metric. But not convergent with discrete metric.

Is there a non-constant convergent sequence with discrete metric ?

• It need not be constant right from the start. Oct 22 '15 at 15:53
• you mean that there is $N$ such that for each $n>N$ , sequence coverges ? That is it has a constant tail after some $N$ . Oct 22 '15 at 15:56
• Yes. In a discrete space, a convergent sequence is eventually constant. But of course it can behave arbitrarily at the beginning. Oct 22 '15 at 15:57

No, in discrete metric, every point is isolated – every singleton $\{x\}$ is open, and hence every sequence converging to $x$ must have a final segment contained in $\{x\}$. So it is eventually constant.
• @AngeloMark Look at the sequence $0,1,1,1,\cdots$ meaning after the initial $0$ all terms are $1$. It converges to $1$ in discrete metric yet is not a constant sequence. Oct 22 '15 at 16:12