When a net is not Cauchy? I have a little problem understanding the concept of Cauchy net, more specifically, when a net is not Cauchy.

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*In a topological linear space, a net $\{x_n:n\in D\}$ ($D$ a directed set) is Cauchy iff for any neighborhood $V$ of $\mathbf{0}$ there is $k\in D$ such that for any $n,m\in D$, $n,m\geq k$ implies $x_n-x_m\in V$.


*Similarly, if $(X,\mathcal{U})$ is a uniform space and $\mathcal{M}:=\{d_\alpha:\alpha\in\mathcal{A}\}$ is a family of metrics on $X$ that generate $\mathcal{U}$, then a net $\{x_n:n\in D\}$ is Cauchy iff for any $\varepsilon>0$ and $\{d_1,\ldots,d_k\}\subset\mathcal{M}$, there is $N\in D$ such that for any $m,n\in D$,
$n,m\geq N$ implies that $\max\limits_{1\leq \ell\leq k}d_\ell(x_n,x_m)<\varepsilon$.
Edit: Based on Andreas Blass observation I made an edit to my question that still covers the situation of the application I have in mind. Here are the edited statements:

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*If a net $\{x_n:n\in D\}$ in a topological linear space $X$ is not a Cauchy net, then  there exists a neighborhood $V$ of $\mathbf{0}$ such that for any $p\in D$, there exists  a nondecreasing sequence $m:\mathbb{N}\rightarrow D$ such that $m_1\geq p$ and $x_{m_{k+1}} - x_{m_k}\in X\setminus V$.


*If a net $\{x_n:n\in D\}$ in a uniform space $(X,\mathcal{U}(\mathcal{M}))$ is not Cauchy, then there exists $\varepsilon>0$, and  finite set $\{d_1,\ldots, d_k\}\subset \mathcal{M}$ such that for any $p\in D$, there exists a nondecreasing sequence $m:\mathbb{N}\rightarrow D$ so that $m_1\geq p$ and $\max\limits_{1\leq \ell\leq k}d_\ell(x_{m_{j+1}},x_{m_j})\geq\varepsilon$.

Initially I had stated that

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*A net $\{x_n:n\in D\}$ in a topological linear space $X$ is not a Cauchy net iff there exists a neighborhood $V$ of $\mathbf{0}$ and a nondecreasing sequence $m:\mathbb{N}\rightarrow D$ such that for any $p\in D$, there is $k_0$ such that $k\geq k_0$ implies $m_k\geq p$ and $x_{m_{k+1}} - x_{m_k}\in X\setminus V$.


*A net $\{x_n:n\in D\}$ in a uniform space $(X,\mathcal{U}(\mathcal{M}))$ is not Cauchy if there exists $\varepsilon>0$, a finite set $\{d_1,\ldots, d_k\}\subset \mathcal{M}$ and a sequence $m:\mathbb{N}\rightarrow D$ such that for any $p\in D$ there is $k_0\in\mathbb{N}$ such that $k\geq k_0$ implies that $m_k\geq p$ and $\max\limits_{1\leq \ell\leq k}d_\ell(x_{m_{j+1}},x_{m_j})\geq\varepsilon$
These initial statements  may be wrong (in the $\Longrightarrow$ direction), although  I have no counterexamples at this point.
 A: The following alternative definition of Cauchy net may be useful:
Lemma: A net $\Phi:=(x_n:n\in D)$ in a topological linear space $X$ is Cauchy iff for any open neighborhood $V$ of $0\in X$, there exists $k_V\in D$ such that
$$ x_n-x_{k_V}\in V,\qquad n\geq k_V$$
A similar statement holds for uniform spaces.
Here is a short proof:
Necessity is obvious. As for sufficiency, let $V$ any open neighborhood of $0\in X$. Let $U$ be a balanced neighborhood of $0\in X$ such that $U+U\subset V$. Let $k_U\in D$ be such that $n\geq k_U$ implies $x_n-x_{k_U}\in U$. Then, for $n,m\geq k_U$
$$x_n-x_m=(x_n-x_{k_U})+(x_{k_U}-x_m)\in U+(-U)=U+U\subset V$$

I am not sure that the statements in the problem hold with "if and only if".
Sufficiency clearly holds, but necessity I'm not so sure  for $D$ may be uncountable.
If the net $\Phi$ is not Cauchy, then there is a neighborhood $V$ of $0\in X$ such that for any $m\in D$, there exists $n\in D$, $n\neq m$ such that $x_n-x_m\in X\setminus V$.   Fix $m_1\geq p\in D$. By the Lemma above, there is $m_2\in D$ such that $m_2>m_1$ and $x_{m_2}-x_{m_1}\in X\setminus V$. Proceeding by induction, suppose  $\{m_1,\ldots, m_k\}\subset D$ have been chosen so  that $m_{j-1}\leq m_j$, $1\leq j\leq k$, and
$x_{m_k}-x_{m_{k-1}}\in X\setminus V$. Another application of the Lemma above gives  $m_{k+1}\in D$ such that $m_{k+1}\geq m_k$, and $x_{m_{k+1}}-x_{m_k}\in X\setminus V$.
From this, I think that the correct "converse statement" for the OP should be:
If $\{x_n:n\in D\}$ is not a Cauchy net, then there exists an open neighborhood $V$ of $0\in X$ such that for any $p\in D$, there is a nondecreasing sequence $m:\mathbb{N}\rightarrow D$ satisfying $m_1\geq p$ and $x_{m_{k+1}}-x_{m_k}\in X\setminus V$.
A similar statement should hold for a net in a uniform space $(X,\mathcal{U}(\mathcal{M}))$.

