Real numbers equipped with the metric $ d (x,y) = |e^x - e^y| $ is an incomplete metric space

Question

How do I show that the real numbers equipped with the metric $ d (x,y) = | e^x - e^y| $ is an incomplete metric space.

If I take $X_n=n$ as non convergent sequence of real numbers. How do I prove that with given metric it is cauchy?

Thanks in advance.

Answer 1

Fix $\varepsilon>0$ and consider the sequence of negative integers. Then $$ d(-n,-m)=|e^{-n}-e^{-m}| < e^{-\min(n,m)} < \epsilon $$ whenever $\min(n,m)$ is sufficiently large. On the other hand, the sequence doesn't converge.