0
$\begingroup$

I need help, I don't know if the following statement is true or false:

"If a sequence $\{x_n\}_{n\in\mathbb{N}} \subset \mathbb{R}$ is divergent, then $\{x_n\}_{n\in\mathbb{N}}$ isn't a Cauchy sequence."

I do know that if $\{x_n\}_{n\in\mathbb{N}} \subset \mathbb{R}$ is convergent, then it is a Cauchy sequence. And this implies that if $\{x_n\}_{n\in\mathbb{N}} \subset \mathbb{R}$ is not a Cauhy sequence then it isn't convergent in $\mathbb{R}$.

Do I have to make distinctions in the statement if the sequence has no limit like $\{(-1)^{n}\}_{n\in\mathbb{N}}$ or if it is no bounded so it diverges to $\pm \infty$?

Thanks in advance.

$\endgroup$
2
$\begingroup$

The real line is complete. Any Cauchy sequence is convergent so any sequence that is not convergent is not Cauchy.

$\endgroup$
1
$\begingroup$

In real line, the double implication holds.

$<x_n>$ is Cauchy $\leftarrow\rightarrow <x_n>$ is convergent

$\endgroup$

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.