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)

\begin{align} \quad (a_{mn})_{m, n=1}^{\infty} = \left ( 2m - n^2 \right )_{m, n = 1}^{\infty} \end{align}

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)

\begin{align} \quad (2m - n^2)_{m, n = 1}^{\infty} & = \begin{pmatrix} a_{11} = 1 & a_{12} = -2 & a_{13} = -7 & \cdots & a_{1n} = 2 - n^2 & \cdots\\ a_{21} = 3 & a_{22} = 0 & a_{23} = - 5 & \cdots & a_{2n} = 4 - n^2 & \cdots\\ a_{31} = 5 & a_{32} = 2 & a_{33} = - 3 & \cdots & a_{3n} = 6 - n^2 & \cdots\\ \vdots & \vdots & \vdots & \ddots & \vdots & \ddots \\ a_{m1} = 2m - 1 & a_{m2} = 2m - 4 & a_{m3} = 2m - 9& \cdots & a_{mn} = 2m - n^2 & \cdots \\ \vdots & \vdots & \vdots & \ddots & \vdots & \ddots \end{pmatrix} \\ & = \begin{pmatrix} 1 & -2 & -7 & \cdots & 2 - n^2& \cdots\\ 3 & 0 & -5 & \cdots & 4-n^2 & \cdots\\ 5 & 2 = & -3 & \cdots & 6 - n^2 & \cdots\\ \vdots & \vdots & \vdots & \ddots & \vdots & \ddots \\ 2m - 1 & 2m - 4 & 2m - 9 & \cdots & 2m - n^2 & \cdots \\ \vdots & \vdots & \vdots & \ddots & \vdots & \ddots \end{pmatrix} \end{align}

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.