Double Sequences of Real Numbers Review

We will now review some of the recent material regarding double sequences of real numbers.

• On the Double Sequences of Real Numbers page we said that a function $f : \mathbb{N} \times \mathbb{N} \to \mathbb{R}$ is said to be a double sequence denoted $(f(m,n))_{m,n=1}^{\infty} = (a_{mn})_{m,n=1}^{\infty}$.

• On the Double Limits and Iterated Limits of Double Sequences of Real Numbers page we said that a double sequence $(a_{mn})_{m,n=1}^{\infty}$ is said to Converge to a limit $A \in \mathbb{R}$ denoted $\displaystyle{\lim_{m, n \to \infty} a_{mn}}$ if for all $\epsilon > 0$ there exists an $N \in \mathbb{N}$ such that if $m, n \geq N$ then:

(1)

\begin{align} \quad \mid a_{mn} - A \mid < \epsilon \end{align}

• If no such $A \in \mathbb{R}$ satisfies the definition above then $(a_{mn})_{m,n=1}^{\infty}$ is said to Diverge.

• We also define the two corresponding Iterated Limits of this double sequence as:

(2)

\begin{align} \quad \lim_{m \to \infty} \left ( \lim_{n \to \infty} a_{mn} \right ) \quad \mathrm{and} \quad \lim_{n \to \infty} \left ( \lim_{m \to \infty} a_{mn} \right ) \end{align}

• We noted that in general, these two limits may not equal each other.

• On the Uniqueness of Double Limits of Double Sequences of Real Numbers page we said that if $(a_{mn})_{m,n=1}^{\infty}$ converges then its double limit is unique, i.e., if $(a_{mn})_{m,n=1}^{\infty}$ converges to both $A$ and $B$ where $A, B \in \mathbb{R}$ then $A = B$.

• We then looked at two nice divergence tests for double sequences on the Divergence Tests for Double Sequences of Real Numbers page.

• We first saw that if $(a_{mn})_{m,n=1}^{\infty}$ is a double sequence that converges to $A \in \mathbb{R}$ if $\displaystyle{\lim_{n \to \infty} a_{mn}}$ exists then $\displaystyle{\lim_{m \to \infty} \left ( \lim_{n \to \infty} a_{mn} \right ) = A}$. Similarly, if $\displaystyle{\lim_{m \to \infty} a_{mn}}$ exists then $\displaystyle{\lim_{n \to \infty} \left ( \lim_{m \to \infty} a_{mn} \right ) = A}$.

• Therefore, if we have that $\displaystyle{\lim_{m \to \infty} \left ( \lim_{n \to \infty} a_{mn} \right ) \neq \lim_{n \to \infty} \left ( \lim_{m \to \infty} a_{mn} \right )}$ we can conclude that the double sequence $(a_{mn})_{m,n=1}^{\infty}$ diverges.

• We also saw that if $(a_{mn})_{m,n=1}^{\infty}$ is a double sequence that converges to $A \in \mathbb{R}$ and if $f, g : \mathbb{N} \to \mathbb{R}$ are two functions such that $\displaystyle{\lim_{t \to \infty} f(t) = \infty}$ and $\displaystyle{\lim_{t \to \infty} g(t) = \infty}$ then:

(3)

\begin{align} \quad \lim_{t \to \infty} a_{f(t),g(t)} = A \end{align}

• Therefore, if there exists functions $f_1, g_1, f_2, g_2 : \mathbb{N} \to \mathbb{R}$ where $\displaystyle{\lim_{t \to \infty} f_1(t), g_1(t), f_2(t), g_2(t) = \infty}$ and such that $\displaystyle{\lim_{t \to \infty} a_{f_1(t), g_1(t)} \neq \lim_{t \to \infty} a_{f_2(t), g_2(t)}}$ then we can conclude that the double sequence $(a_{mn})_{m,n=1}^{\infty}$ diverges.

• On the Boundedness of Double Sequences of Real Numbers we said that a double sequence $(a_{mn})_{m,n=1}^{\infty}$ is Bounded Above if there exists an $M' \in \mathbb{R}$ such that for all $m, n \in \mathbb{N}$ we have that:

(4)

\begin{align} \quad a_{mn} \leq M' \end{align}

• Similarly, we said that a double sequence $(a_{mn})_{m,n=1}^{\infty}$ is Bounded Below if there exists an $m' \in \mathbb{R}$ such that for all $m, n \in \mathbb{N}$ we have that:

(5)

\begin{align} \quad m' \leq a_{mn} \end{align}

• Furthermore, a double sequence $(a_{mn})_{m,n=1}^{\infty}$ is said to be Bounded if it is both bounded above and below, i.e., if there exists an $M^* \in \mathbb{R}$, $M^* > 0$ such that for all $m, n \in \mathbb{N}$ we have that:

(6)

\begin{align} \quad \mid a_{mn} \mid \leq M^* \end{align}

• A double sequence that is not bounded is said to be Unbounded.

• On the The Boundedness of Convergent Double Sequences of Real Numbers we saw a familiar theorem translated for double sequences. We proved that if $(a_{mn})_{m,n=1}^{\infty}$ is a convergent double sequence that $(a_{mn})_{m,n=1}^{\infty}$ is also bounded.

• On the The Squeeze Theorem for Double Sequences of Real Numbers we proved the Squeeze Theorem for double sequences of real numbers. We saw that if $(a_{mn})_{m,n=1}^{\infty}$, $(b_{mn})_{m,n=1}^{\infty}$, and $(c_{mn})_{m,n=1}^{\infty}$ are double sequences such that there exists an $M \in \mathbb{N}$ such that $a_{mn} \leq b_{mn} \leq c_{mn}$ for all $m, n \geq N$, and if $(a_{mn})_{m,n=1}^{\infty}$ and $(c_{mn})_{m,n=1}^{\infty}$ converge to $L$ then $(b_{mn})_{m,n=1}^{\infty}$ also converges to $L$.

• On the Cauchy Convergence Criterion for Double Sequences we said that a double sequence $(a_{mn})_{m,n=1}^{\infty}$ is called a Cauchy Double Sequence (or simply, "Cauchy") if for all $\epsilon > 0$ there exists an $N \in \mathbb{N}$ such that if $m_1, n_1, m_2, n_2 \geq N$ then:

(7)

\begin{align} \quad \mid a_{m_1n_1} - a_{m_2n_2} \mid < \epsilon \end{align}

• We then proved that a double sequence $(a_{mn})_{m,n=1}^{\infty}$ converges if and only if it is Cauchy which is an extension to the analogous theorem for regular sequences of real numbers.