1
$\begingroup$

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

$\endgroup$
3
$\begingroup$

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 $d_2$.

| cite | improve this answer | |
$\endgroup$
0
$\begingroup$

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

| cite | improve this answer | |
$\endgroup$
  • 2
    $\begingroup$ It's worth mentioning that in both cases, the answer is "no, it's not complete." $\endgroup$ – user296602 Dec 27 '15 at 8:18

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.