Let $S_n=\sum_{k=1}^n \frac{1}{k}$ the $n$-th partial sum in the harmonic series. Once can prove that
$$
\lim_{n\rightarrow\infty}|S_{n+p}-S_n|=0,
$$
for all $p\in\mathbb{N}$. On the other hand, the harmonic series diverges. So the question is why the above limit "fulfills" the Cauchy criterion for series, but the harmonic series still diverges?
The Cauchy Criterion states that for a sequence $(a_n)$ in the complex numbers, $\sum_n a_n$ converges if and only if to each $\epsilon>0$ there is a $n_0\in\mathbb{N}$ such that $q>p\geq n_0$ implies $|\sum_{n=p+1}^q a_n|=|S_q-S_p|<\epsilon$.
Best Answer
You know that for all $p$ and $\varepsilon$ there is some $N_{p,\varepsilon}$ such that, for all $n\ge N_{p,\varepsilon}$, $\lvert a_{n+p}-a_n\rvert\le \varepsilon$.
What you do not know is wheter or not for all $\varepsilon$ there is some $N_\varepsilon$ such that, for all $n\ge N_\varepsilon$ and for all $p$, $\lvert a_{n+p}-a_n\rvert\le\varepsilon$.
As of why these need not be equivalent, the instance at hand shows.