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Suppose I consider a triple sequence indexed by $l,m,n$ and I take limits in the order of $l,m,$ and $n$. Then, should I write this sequence as $x_{l,m,n}$ or $x_{n,m,l}$?

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It doesn't matter. – Thomas Andrews Jan 16 '12 at 1:23
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(just not to leave this question "unanswered"), Thomas' comment is correct: it doesn't matter. The order of taking the limits, on the other hand, does matter, and the statement "in the order of $l$, $m$ and $n$" is ambiguous.

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The order of the indices of $x$ matters so that the reader can understand which is which. As Thomas Andrews says, it does not matter which order this is so long as you can correctly identify the meaning.

What may matter in some cases is the order in which the limits are wrtten, so conventionally you want to write something like $$\lim_{n \to \infty}\, \lim_{m \to \infty}\, \lim_{l \to \infty} \, x$$ if you want to take the limit with respect to $l$ before taking the limit with respect to $m$ before taking the limit with respect to $n$.

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