[Math] A sequence in $\mathbb{R}$ that has no Cauchy subsequence

Question

Give an example of a sequence in $\mathbb{R}$ which has no subsequence which is a Cauchy sequence.

I can find out a sequence that is not a Cauchy sequence such as $\{\ln(n)\}$ once $|\ln(n)-\ln(n+1)|=0$ but $|\ln(n)-\ln(2n)|=|\ln(\frac{1}{2})|>\epsilon$
$\forall \epsilon<\ln(\frac{1}{2})$

I can still find a subsequence of the type ${\ln(2n)}_{2n\in\mathbb{N}}$ such that $|\ln(2n)-\ln(2n+1)|=0$

Question:

What should I do to get a sequence that has no Cauchy subsequence?

Thanks in advance!

Answer 1

Take $a_n=n$. Then for any subsequence $n_k$, $|n_k-n_{k-1}|\geq 1$. So, it has no Cauchy sub-sequence.