---
tag: B70G8FYPG0
kind: concept
name: Double sequence
msc: [40B05]
sources:
  - url: https://en.wikipedia.org/wiki/Iterated_limit
---

## Summary

A double sequence is a family $(x_{m,n})$ indexed by pairs of natural numbers, like an infinite matrix.
It has several notions of limit.
There is the joint limit as $m$ and $n$ both grow, and there are two iterated limits taken one index at a time.
These limits can differ, and theorems such as the [[Moore–Osgood theorem|QN2210XCST]] say when they agree.
The exact definition is in [[double sequence|WK5V372TG3]].

## Explanations

### L1 human

A double sequence is an infinite table of numbers with rows and columns, one entry for each pair $(m, n)$.
For example, $x_{m,n} = m + n$ gives the table with rows $0, 1, 2, \ldots$ and $1, 2, 3, \ldots$ and so on.
One can let the row number grow first, or the column number first, or both together, and the outcomes need not be the same.