Complete Metric Spaces

Question

Consider $\mathbb{R}$ with the following metrics

$d_1(x, y) = |e^x - e^y|,\,\forall{x, y} \in \mathbb{R};$

$d_2(x, y) = |\tan^{-1}(x)-\tan^{-1}(y)|,\,\forall{x, y} \in \mathbb{R}$

Are the metric spaces $(\mathbb{R}, d_1)$ and $(\mathbb{R}, d_2)$ complete?

I need to prove that all sequences converge to the metric spaces in order to prove that they are complete. Am I right? I'm not sure how to work with a completely arbitrary sequence in order to prove that it holds for all sequences. Any help will be appreciated. Thanks

Answer 1

You need to be much more precise with your language: The statement

I need to prove that all sequences converge to the metric spaces in order to prove that they are complete.

unfortunately doesn't make any sense. Completeness is the property that any Cauchy sequence in the space converges to a limit in the space. So I'd suggest that you start with a Cauchy sequence and a putative limit, and hope that your sequence converges to the limit and that the limit is in the space.

For example: Take the sequence $-1, -2, -3, ...$. Do you see why it's Cauchy with respect to the metric $d_1$? Do you see that it is divergent the space $(\mathbb{R}, d_1)$?

A similar method can be used to study $d_2$.

Answer 2

You have to show that an arbitrary Cauchy sequence conveges or not.