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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.

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    What precisely is meant by $\lim_{(x,y)\to(\infty,\infty)}f(x,y)=L$, here?2012-05-13
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    Yes, assuming that your definition of $\lim\limits_{(j,k)\to(\infty,\infty)}f(j,k)=L$ is that for each $\epsilon>0$ there is $n_0\in\Bbb N$ such that $|f(j,k)-L|<\epsilon$ whenever $j,k\ge n_0$.2012-05-13
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    Thanks, that's what I meant.2012-05-14

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