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 $|p_n – p_m|$?).
Best Answer
The cauchy criterion states $\forall \epsilon>0 \exists N\in\mathbb{N}\forall n\geq N \forall m\geq N(|p_n-p_m|)<\epsilon)$. The negation should be $\exists \epsilon > 0 \forall N\in \mathbb{N} \exists n\geq N \exists m\geq N (|p_n -p_m|\ge\epsilon)$.
I think your mistake can be attributed to the following: the cauchy criterion is often written $\forall \epsilon>0 \exists N\in\mathbb{N}\forall n,m\geq N (|p_n-p_m|)<\epsilon)$, which people pronounce as "for all epsilon greater than zero, there existn $N$ in $\mathbb{N}$ such that, for all $n$ and $m$ greater than $N$, $|p_n-p_m|<\epsilon$". However, the "$n$ and $m$" is misleading. This does not mean logical and, but rather just suggest you quantify over $n$ and you then quantify over $m$.