Cauchy property and limit in two variables

Question

Let $(x_{n})_{n\in\mathbb{N}}$ be a sequence of real numbers with the property that \begin{equation} \forall\varepsilon>0\quad\exists N\in\mathbb{N}\quad\forall\color{red}m \color{red}>\color{red}n\geq N:\quad\lvert x_{n}-x_{m}\lvert<\varepsilon. \end{equation} Given the above, I would like to understand what the expression \begin{equation} \lim_{\substack{m,n\to\infty\\ m>n}}\lvert x_{n}-x_{m}\rvert=0 \end{equation} exactly means and why this ''Cauchy property'' suffices for $(x_{n})_{n\in\mathbb{N}}$ to admit a limit?

Answer 1

A double sequence $a_{m,n}$ is formally a function $\mathbb{N}\times\mathbb{N}\to\mathbb{R}$. We say $\lim\limits_{m,n\to\infty} a_{m,n}=a$ if:

$\forall\epsilon>0 \ \ \exists n_0\in\mathbb{N} \ \ \forall m,n\geq n_0: |a_{m,n}-a|<\epsilon$.

So this is a natural generalization of a regular limit. The condition that $(x_n)$ is Cauchy exactly means that $\lim\limits_{m,n\to\infty} |x_m-x_n|=0$, this is immediate from the definition.