What is the negation of the Cauchy criterion for sequences? I initially believed that the negation of the Cauchy criterion for a sequence ${p_n}$ in $\mathbb{R}$ is
$\exists \epsilon>0:\forall N\in \mathbb{N}, \exists (n\geq N \lor m\geq N):|p_n - p_m| \geq \epsilon$
due to the effect of DeMorgan's Law on the part $\forall(n \geq N \land m\geq N)$ ... of the Cauchy criterion statement.
However, I am unsure if this negation is correct (if, for example, there only exists $n\geq N : |p_n - p_m| \geq \epsilon$, what is the significance of $m$ in $|p_n - p_m|$?).