Definitions: If $f$ is a function from a set $A$ (not necessarily countable) into a Banach space $V$, we say that the series of $f$ over $A$ converges if there exists an element $v \in V$ such that for all $\epsilon>0$, there exists a finite subset $B$ of $A$ such that, for all finite subset $C$ of $A$, if $B$ is contained in $C$, than the distance between the sum over $C$ of $f$ and $v$ is less then $\epsilon$.
If $f$ is a function from a set $A$ into a Banach space $V$, we say that the series of $f$ over $A$ is Cauchy if for all $\epsilon>0$, there exists a finite subset $B$ of $A$ such that, for all finite subsets $C$ and $D$ of $A$, if $B$ is contained in $C$ and $D$, then we have that the distance between the sum over $C$ of $f$ and the sum over $D$ of $f$ is less then $\epsilon$.
Question: is it true that a Cauchy series is convergent? Proof? Counterexamples?