I have recently just come across the definition of net and directed set. One thing I noticed is that the criterion for a sequence to be Cauchy can be formulated in term of net. Recall the definition of Cauchy sequence in a metric space $X$:

A sequence $(a_n)$ in $X$ is called a Cauchy sequence if for any $\epsilon>0$ there exists $N\in \Bbb N$ such that $d(a_n,a_m)<\epsilon$ whenever $n,m\ge N$.

Let $\Bbb N^2$ be a directed set with respect to the relation $\prec$, defined by $(m_1,n_1)\prec (m_2,n_2)$ whenever $m_1\le m_2$ and $n_1\le n_2$. Then the net $A:\Bbb N^2\to\Bbb R$ associated with the sequence $(a_n)$ defined by $$ A_{m,n}=d(a_m,a_n) $$ can provide an alternative to the usual criterion

A sequence $(a_n)$ in $X$ is called a Cauchy sequence if its associated net $A_{m,n}$ converges toward $0$, i.e. $\lim A_{m,n}=0$.

I suppose that this point of view is already well known to the learned. I think this is an interesting way to view a Cauchy sequence but is there any real advantages of looking at it this way? I mean it's nice and all but do we gain any new insight into the Cauchy sequence?

One good thing I can think of is that this make the informal statement "If $(a_n)$ is a Cauchy sequence then $\lim_{m,n\to\infty}|a_m-a_n|=0$" somewhat more rigorous.

However, I can see a disadvantage (?) of this definition. According to the standard proof of the fact that a Cauchy sequence $(f_n)$ of bounded continuous functions on $X$ (with supremum norm) converges uniformly to its pointwise limit $f$, there is this step:

Let $N\in\Bbb N$ be a number such that $|f_n(x)-f_m(x)|<\epsilon$ for all $x\in X$ and $n,m\ge N$. Then $|f(x)-f_N(x)|=\lim_{n\to \infty}|f_n(x)-f_N(x)|\le\epsilon$ for all $x\in X$ etc. etc.

where we have to take the limit of only one of the $a_m,a_n$, our definition using directed set doesn't seem to provide us with a way to do that. Is there a way to resolve this using only the net definition without going back to the classic definition of Cauchy sequence?


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