# Thread: Non-convergent sequences in metric spaces

1. ## Non-convergent sequences in metric spaces

I need to prove that in any metric space containing more than one point exists a non-convergent sequence. This seems easy enough, but after wrestling with definitions of the metric space and the limit of a sequence I can't seem to produce a proof. Do I need to know more to prove this?

2. Originally Posted by iligabor
I need to prove that in any metric space containing more than one point exists a non-convergent sequence. This seems easy enough, but after wrestling with definitions of the metric space and the limit of a sequence I can't seem to produce a proof. Do I need to know more to prove this?

Let x,y, be two different points in a metric space. What about the seq. $x_n=\left\{\begin{array}{ll}x&if\,\,n\,\,is\,\,odd \\y&if\,\,n\,\,is\,\,even\end{array}\right.$ ?

Tonio

3. Thank you, now I see it.