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

$\displaystyle (X,d)$ is metric space. for sequence $\displaystyle (x_n)_{n\in \mathbb{N}} \subset X$ we say that is Cauchy sequence if

$\displaystyle (\forall \varepsilon >0) (\exists n_0=n_0(\varepsilon) \in \mathbb{N}) (\forall n,m \in \mathbb{N}) (n,m\ge n_0 \Rightarrow d(x_n,x_m) < \varepsilon )$

meaning that, sequence is Cauchy sequence if

$\displaystyle \displaystyle \lim _{n,m \to \infty } d(x_n,x_m) = 0 $

Is it sufficient to show just

$\displaystyle \displaystyle \lim _{n,m \to \infty } d(x_n,x_m) = 0 $

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 $\displaystyle (R,d) $ is not complete. I have created such a metric (i think) with

$\displaystyle d(x,y)= |exp(-x) - exp(-y)| $ but i am struggling to show with the epsilon definition that the sequence of natural numbers is cauchy on it.