---
tag: WK5V372TG3
kind: definition
name: Double sequence
msc: [40B05]
defines: B70G8FYPG0
uses: [0QE8STGJH2, 0EPAT51DXQ]
sources:
  - url: https://en.wikipedia.org/wiki/Iterated_limit
---

## Definition

Let $X$ be a set.
A double sequence in $X$ is a [[function|0EPAT51DXQ]] $x \colon \N \times \N \to X$.
Its value at $(m, n)$ is written $x_{m,n}$, and the double sequence is written $(x_{m,n})_{m, n \in \N}$.

## Remarks

A double sequence is a [[sequence|0QE8STGJH2]] indexed by $\N \times \N$ instead of $\N$.
Example: $x_{m,n} = m/(m + n + 1)$ is a double sequence of real numbers.
For fixed $m$ it tends to $0$ as $n \to \infty$, and for fixed $n$ it tends to $1$ as $m \to \infty$.
So $\lim_m \lim_n x_{m,n} = 0$ and $\lim_n \lim_m x_{m,n} = 1$: the two iterated limits differ.
Non-example: a triangular array $(x_{m,n})_{0 \le n \le m}$ is not a double sequence, since it is defined only for $n \le m$; it becomes one after extension by $0$.
The joint limit in the sense of [[Pringsheim convergence|QYTGCW1GTK]] is the usual notion of limit for double sequences.

## Explanations

### L1 human

A double sequence assigns an element to each pair $(m, n)$ of natural numbers, like an infinite matrix.