How to state that a sequence is Cauchy in terms of $\limsup$ and $\liminf$?

How to state that a sequence is Cauchy in terms of $\limsup$ and $\liminf$?

For example, is it true that a sequence $(a_n)_{n=1}^{\infty}$ is Cauchy iff $\displaystyle\limsup_{n\to\infty}|a_{n+k}-a_n|=0$ for all $k\in\mathbb{N}$?

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The condition that you state is not equivalent to being Cauchy. To see this, consider
$$
a_n =\sum_{t =1}^n \frac {1}{t}.
$$

Then
$$
0\leq a_{n+k} – a_n =\sum_{t = n+1}^{n+k} \frac {1}{t}\leq \sum_{t = n+1}^{n+k}\frac {1}{n}\leq \frac{k}{n}\to 0
$$
as $n\to\infty $ for every $k $, but as is well known, we have $a_n \to \infty $, so that the sequence is not Cauchy.

As noted by @Börge, a real sequence $(a_n)_n $ is Cauchy if and only if it is convergent if and only if $\limsup_n a_n = \liminf_n a_n \in \Bbb {R} $.

Another way would be to require
$$
a_{n+k}-a_n \to 0
$$
for $n \to \infty $, uniformly with respect to $k $.