I know the origin in analysis as it is derived from our good old chap Cauchy himself. However I have read about Cauchy sequences in algebra, I'll use groups for this one, let $G$ be a group and $G_i\triangleleft G$ for all $i\in\mathbb{N}$ along with $G_0=G$ and our filtration is then
$$G_0\supseteq G_1 \supseteq G_1\supseteq\ldots$$ Next we let $\{x_i\}$ be a sequence, it is a cauchy sequence if and only if we have that for a given $m\in\mathbb{N}$ there exists an $N$ such that $\forall i,j > N$ we have that $x_i x_j^{-1}\in G_m$.
I think I got that right from memory, anyhow I am curious as to why this is called a cauchy sequence? While I understand there are definitional simularities in how it is phrased, the whole "Find $N$ such that $i,j>N$" bit is right from the traditional real analysis definition. I can even see the parallel between $\epsilon$ in the analysis definition and our $m$ here which really marks a normal subgroup. In this chain we have that the "size" of the groups get's smaller (for simplicity sake assuming it doesn't stagnate), so $m\to\infty$ would be a parallel to $\epsilon\to 0$. And of course $x_i x_j^{-1}\in G_m$ is parallel to $|x_i-x_j|<\epsilon$.
Is this the reason why it is called cauchy because of these striking similarities? Where did this name come from? Who came with it for algebra?