Double Limits and Iterated Limits of Double Seqs. of Real Numbers

# Double Limits and Iterated Limits of Double Sequences of Real Numbers

Recall from the Double Sequences of Real Numbers page that a function $f : \mathbb{N} \times \mathbb{N} \to \mathbb{R}$ is called a double sequence denoted $(a_{mn})_{m, n=1}^{\infty}$ where $f(m, n) = a_{mn}$ for all $m, n \in \mathbb{N}$.

Like with regular sequences of real numbers, we can define when a double sequence converges or not. We first define a double limit of a double sequence.

 Definition: The double sequence $(a_{m, n})_{m, n = 1}^{\infty}$ is said to Converge to the real number $A \in \mathbb{R}$ if for all $\epsilon > 0$ there exists an $N \in \mathbb{N}$ such that if $m, n \geq N$ then $\mid a_{m, n} - A \mid < \epsilon$ and we say $A$ is the Double Limit of this double sequence written $\displaystyle{\lim_{m, n \to \infty} a_{m, n} = A}$. If no such $A \in \mathbb{R}$ satisfies this, then we say that the the double sequence $(a_{m, n})_{m, n = 1}^{\infty}$ diverges.

In general, it may be quite a bit more cumbersome to show that a double sequence converges to a particular limit $A$, and even if a double sequence does converge, it may be difficult to guess the value $A$ that it converges to.

Before we move on, we will also need to describe another type of limit of a double sequence known as an iterated limit.

 Definition: Let $(a_{m, n})_{m, n=1}^{\infty}$ be a double sequence. Then the Iterated Limits of this double sequence are defined as $\displaystyle{\lim_{m \to \infty} \left ( \lim_{n \to \infty} a_{m, n} \right )}$ and $\displaystyle{\lim_{n \to \infty} \left ( \lim_{m \to \infty} a_{m, n} \right )}$.

It is very important to note the distinction between the double limit and the iterated limits of a double sequence. In general, these three limits may be different.

For example, consider the following double sequence:

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

The two iterated limits for this double sequence are:

(2)
\begin{align} \quad \lim_{m \to \infty} \left ( \lim_{n \to \infty} \frac{m^2 + 2n^2}{n^2} \right ) = \lim_{m \to \infty} \left ( \lim_{n \to \infty} \frac{m^2}{n^2} + 2 \right ) = \lim_{m \to \infty} 2 = 0 \end{align}
(3)
\begin{align} \quad \lim_{n \to \infty} \left ( \lim_{m \to \infty} \frac{m^2 + 2n^2}{n^2} \right ) = \infty \end{align}