A sequence of certain elements numbered by two indices:
As compared to ordinary sequences (i.e. sequences numbered with the aid of one index), double sequences have a number of distinguishing features; thus, there are several definitions of the limit of a double sequence which are not mutually equivalent.
The concept of a numerical double sequence is closely connected with that of a numerical double series
both the terms and the (rectangular) partial sums of such a double series,
constitute a double sequence. See also Double series.
A double sequence for which the notion of its limit causes trouble is given by, e.g.,
Letting $m\to\infty$ for $n$ fixed, and subsequently $n\to\infty$ gives as limit 1. On the other hand, letting $n\to\infty$ for $m$ fixed, and subsequently $m\to\infty$ gives 0.
Double sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Double_sequence&oldid=32337