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

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 $$\pm \infty$$?

$$$$ is Cauchy $$\leftarrow\rightarrow $$ is convergent