I am working on a problem and was just reading a thread where a user posted :

is metric space. for sequence we say that is Cauchy sequence if

meaning that, sequence is Cauchy sequence if

Is it sufficient to show just

for a sequence to be cauchy? Or should it be done using the epsilon definition (which i am struggling with).

Basically i'm trying to construct a metric on the real numbers so that is not complete. I have created such a metric (i think) with

but i am struggling to show with the epsilon definition that the sequence of natural numbers is cauchy on it.