If a sequence is divergent in $\mathbb{R}$ , then it isn't a Cauchy sequence

Question

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.

Answer 1

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

Answer 2

In real line, the double implication holds.

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