Is it possible to define a Cauchy sequence as follows?
Let $(X,d)$ be a metric space and $(x_{n})_{n\in \mathbb{N}}$ be a sequence in it. Then $(x_{n})_{n\in \mathbb{N}}$ is Cauchy iff $\lim_{(j,k)\to (\infty,\infty) }d(x_{j},x_{k})=0$.
Thanks.
Note: By $\lim_{(j,k)\to (\infty,\infty) }d(x_{j},x_{k})=0$, I mean the standard definition of the limit for a function $d:\mathbb{R}\times \mathbb{R} \to \mathbb{R}$ using Euclidean metric $ d_{E}$.
Note2: Thanks for the answers. Conclusion: It can be defined as a double limit indeed, as indicated by http://books.google.co.uk/books?id=lZU0CAH4RccC&pg=PA34&redir_esc=y#v=onepage&q&f=false page 33.