A sequence that satisfies the condition: $\forall p \in \mathbb{N}: \lim_\limits{n \to \infty } | a_p - a_{p+1}| = 0$ but is not a cauchy sequence.
$\lim_{n\to\infty} a_n=a$ if and only if $\forall p\in \Bbb N$, $\lim_{n\to\infty} |a_{n+p}-a_n|=0$