# Double Sequences of Real Numbers

Awhile ago, we defined a sequence of real numbers to be a function $f : \mathbb{N} \to \mathbb{R}$ denoted $(a_n)_{n=1}^{\infty}$ where $f(n) = a_n$ for each $n \in \mathbb{N}$ (or more generally, we can start a sequence at any integer as opposed to $1$).

We can define a new type of sequence called a double sequence which we define below.

Definition: Let $f : \mathbb{N} \times \mathbb{N} \to \mathbb{R}$. Then $f$ is said to be a double sequence denoted $(a_{mn})_{m, n = 1}^{\infty}$. The term $a_{mn}$ is called the General Term of the double sequence and $f(m, n) = a_{mn}$ for all $m, n \in \mathbb{N}$. |

*We can also define double sequences as functions $f : \mathbb{Z} \times \mathbb{Z} \to \mathbb{R}$ but for convenience we will assume that the starting indices $m$ and $n$ are $1$.*

For example, consider the following double sequence $(a_{mn})_{m, n = 1}^{\infty}$:

(1)Visually, we can represent the terms in a general sequence as an infinite array where the entry in row $m$ column $n$ is the term $a_{mn}$. For example:

(2)Notice that in a double sequence that for every fixed row $m$, $(a_{mn})_{n=1}^{\infty}$ is a row sequence of real numbers. Similarly, for every fixed column $n$, $(a_{mn})_{m=1}^{\infty}$ is a column sequence of real numbers.