Cauchy equivalent condition 
Let $E$ be a topologic vector space. $x_i$ with $i\in I$ is a Cauchy net iff $\lim_j \ x_{\phi(j)}-x_{\psi(j)}=0$ for any $(\phi,\psi)$ pair of cofinal and increasing maps from $J$ to $I$.

I proved the necessity part but I struggle with sufficiency. I assumed the given limit is zero but $x_i$ is not a Cauchy net. Then $\exists V$ $0$-neighborhood $\forall i\in I \ \ \exists i_1,i_2 \geq i$ such that $x_{i_1}-x_{i_2} \notin V$. Also given limit says $\forall V$ $0$-neighborhood  $\exists j_0 \in J$ such that $\forall j\geq j_0 \ \ x_{\phi(j)}-x_{\psi(j)}\in V$
I cannot see the relation between $i_1,i_2$ and maps for getting contradiction because I am not sure about surjectivity. How can I proceed to get given limit is not zero?
I appreciate any kind of help.
Thanks
 A: This is a similar question as Equivalent condition for Cauchy Sequences.
We have to prove that if the limit-condition is satisfied, then $(x_i)$ is a Cauchy net.  Let us do it by contraposition.
So assume that $(x_i)$ is not a Cauchy net. This means that there exists an open neigborhood $V$ of $0$ such that for each $i \in I$ there exist $\phi(i), \psi(i) \ge i$ such that $x_{\phi(i)} - x_{\psi(i)} \notin V$. This gives us functions $\phi, \psi : I \to I$ which are cofinal  and we have $x_{\phi(i)} - x_{\psi(i)} \notin V$, thus $\lim_{i \in I}(x_{\phi(i)} - x_{\psi(i)}) \ne 0$. Unfortunately there is no reason why these functions should be increasing.
What can be done to get increasing cofinal functions on some $J$?
Let $J $ be the set of all finite non-empty subsets of $I$, ordered by inclusion. This is a directed set. Note that $I$ can be canonically identified with the subset $I' \subset J$ of one-element subsets ($i \equiv \{i\}$), but the original order of $I$ is forgotten in this identification. Regarding the above $\phi, \psi$ as functions $I' \to I$, we shall extend them to functions $J \to I$. The construction will be performed by induction on the number $n$ of elements of $S \in J$.
$n=1$: If $S = \{i\}$, take $\phi(S) = \phi(i),  \psi(S) =  \psi(i)$.
Assume $\phi(S), \psi(S)$ have been defined for all $S$ with $\le n$ elements such that

*

*$\phi(S') \le \phi(S)$ and $\psi(S') \le \psi(S)$ if $S' \subset S$

*$\phi(S) \ge i$ and $\psi(S) \ge i$ for all $i \in S$

*$x_{\phi(S)} - x_{\psi(S)} \notin V$
To define $\phi(T), \psi(T)$ for a set $T$ with $n+1$ elements we consider the set of all subsets $S_1,\ldots, S_{n+1} \subset T $ having $n$ elements. Since $I$ is directed, we find $k \in I$ such that $k \ge i$ for all $i \in T$ and $k \ge \phi(S_r), \psi(S_r)$ for $r = 1,\ldots, n+1$. Define $\phi(T) = \phi(k), \psi(T) = \psi(k)$. This definitions assures that 1. - 3. are satisfied.
This construction yields increasing functions (condition 1.) which are also cofinal (they are even cofinal on the subset $J_1$). Clearly 3. implies  $\lim_{S \in J}(x_{\phi(S)} - x_{\psi(S)}) \ne 0$.
