Dear Colleagues,
Could you please help me in solving the following problem:
If is a Cauchy sequence in the metric space and in such that as then show that is Cauchy in .
Have you tried drawing a 2d diagram? (they can be great models for metric spaces) Remember since the 's are Cauchy you can find an epsilon ball that the tail never leaves.
Now since the 's get close to the 's can find another N such that this distance also falls in another epsilon ball.
As I said draw a picture and what fraction of epsilon you need.
Hint you will need 3 balls.