Sequence of more than one index

 

Sequence of more than one index[edit]

Sometimes one may also consider a sequence with more than one index, for example, a double sequence ${\displaystyle ({�}_{�,�})}$. This sequence has a limit ${\displaystyle �}$ if it becomes closer and closer to ${\displaystyle �}$ when both n and m becomes very large.

Example[edit]

• If ${\displaystyle {�}_{�,�}=�}$ for constant c, then ${\displaystyle {�}_{�,�}\to �}$.
• If ${\displaystyle {�}_{�,�}=\frac{1}{�+�}}$, then ${\displaystyle {�}_{�,�}\to 0}$.
• If ${\displaystyle {�}_{�,�}=\frac{�}{�+�}}$, then the limit does not exist. Depending on the relative "growing speed" of n and m, this sequence can get closer to any value between 0 and 1.

Definition[edit]

We call ${\displaystyle �}$ the double limit of the sequence ${\displaystyle ({�}_{�,�})}$, written

${\displaystyle {�}_{�,�}\to �}$, or

${\displaystyle \underset{\begin{array}{c}�\to \mathrm{\infty }\\ �\to \mathrm{\infty }\end{array}}{lim}{�}_{�,�}=�}$,

if the following condition holds:

For each real number , there exists a natural number ${\displaystyle �}$ such that, for every pair of natural numbers ${\displaystyle �,�\ge �}$, we have .^[10]

In other words, for every measure of closeness ${\displaystyle �}$, the sequence's terms are eventually that close to the limit. The sequence ${\displaystyle ({�}_{�,�})}$ is said to converge to or tend to the limit ${\displaystyle �}$.

Symbolically, this is:

.

Note that the double limit is different from taking limit in n first, and then in m. The latter is known as iterated limit. Given that both the double limit and the iterated limit exists, they have the same value. However, it is possible that one of them exist but the other does not.

Infinite limits[edit]

A sequence ${\displaystyle ({�}_{�,�})}$ is said to tend to infinity, written

${\displaystyle {�}_{�,�}\to \mathrm{\infty }}$, or

${\displaystyle \underset{\begin{array}{c}�\to \mathrm{\infty }\\ �\to \mathrm{\infty }\end{array}}{lim}{�}_{�,�}=\mathrm{\infty }}$,

if the following holds:

For every real number ${\displaystyle �}$, there is a natural number ${\displaystyle �}$ such that for every pair of natural numbers ${\displaystyle �,�\ge �}$, we have ; that is, the sequence terms are eventually larger than any fixed ${\displaystyle �}$.

Symbolically, this is:

.

Similarly, a sequence ${\displaystyle ({�}_{�,�})}$ tends to minus infinity, written

${\displaystyle {�}_{�,�}\to -\mathrm{\infty }}$, or

${\displaystyle \underset{\begin{array}{c}�\to \mathrm{\infty }\\ �\to \mathrm{\infty }\end{array}}{lim}{�}_{�,�}=-\mathrm{\infty }}$,

if the following holds:

For every real number ${\displaystyle �}$, there is a natural number ${\displaystyle �}$ such that for every pair of natural numbers ${\displaystyle �,�\ge �}$, we have ; that is, the sequence terms are eventually smaller than any fixed ${\displaystyle �}$.

Symbolically, this is:

.

If a sequence tends to infinity or minus infinity, then it is divergent. However, a divergent sequence need not tend to plus or minus infinity, and the sequence ${\displaystyle {�}_{�,�}=(-1{)}^{�+�}}$ provides one such example.

Pointwise limits and uniform limits[edit]

For a double sequence ${\displaystyle ({�}_{�,�})}$, we may take limit in one of the indices, say, ${\displaystyle �\to \mathrm{\infty }}$, to obtain a single sequence ${\displaystyle ({�}_{�})}$. In fact, there are two possible meanings when taking this limit. The first one is called pointwise limit, denoted

${\displaystyle {�}_{�,�}\to {�}_{�}\text{pointwise}}$, or

${\displaystyle \underset{�\to \mathrm{\infty }}{lim}{�}_{�,�}={�}_{�}\text{pointwise}}$,

which means:

For each real number  and each fixed natural number ${\displaystyle �}$, there exists a natural number  such that, for every natural number ${\displaystyle �\ge �}$, we have .^[11]

Symbolically, this is:

.

When such a limit exists, we say the sequence ${\displaystyle ({�}_{�,�})}$ converges pointwise to ${\displaystyle ({�}_{�})}$.

The second one is called uniform limit, denoted

${\displaystyle {�}_{�,�}\to {�}_{�}\text{uniformly}}$,

${\displaystyle \underset{�\to \mathrm{\infty }}{lim}{�}_{�,�}={�}_{�}\text{uniformly}}$,

${\displaystyle {�}_{�,�}\rightrightarrows {�}_{�}}$, or

${\displaystyle \underset{�\to \mathrm{\infty }}{\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{f}lim}{�}_{�,�}={�}_{�}}$,

which means:

For each real number , there exists a natural number  such that, for every natural number ${\displaystyle �}$ and for every natural number ${\displaystyle �\ge �}$, we have .^[11]

Symbolically, this is:

.

In this definition, the choice of ${\displaystyle �}$ is independent of ${\displaystyle �}$. In other words, the choice of ${\displaystyle �}$ is uniformly applicable to all natural numbers ${\displaystyle �}$. Hence, one can easily see that uniform convergence is a stronger property than pointwise convergence: the existence of uniform limit implies the existence and equality of pointwise limit:

If ${\displaystyle {�}_{�,�}\to {�}_{�}}$ uniformly, then ${\displaystyle {�}_{�,�}\to {�}_{�}}$ pointwise.

When such a limit exists, we say the sequence ${\displaystyle ({�}_{�,�})}$ converges uniformly to ${\displaystyle ({�}_{�})}$.

Iterated limit[edit]

For a double sequence ${\displaystyle ({�}_{�,�})}$, we may take limit in one of the indices, say, ${\displaystyle �\to \mathrm{\infty }}$, to obtain a single sequence ${\displaystyle ({�}_{�})}$, and then take limit in the other index, namely ${\displaystyle �\to \mathrm{\infty }}$, to get a number ${\displaystyle �}$. Symbolically,

${\displaystyle \underset{�\to \mathrm{\infty }}{lim}\underset{�\to \mathrm{\infty }}{lim}{�}_{�,�}=\underset{�\to \mathrm{\infty }}{lim}{�}_{�}=�}$.

This limit is known as iterated limit of the double sequence. Note that the order of taking limits may affect the result, i.e.,

${\displaystyle \underset{�\to \mathrm{\infty }}{lim}\underset{�\to \mathrm{\infty }}{lim}{�}_{�,�}\ne \underset{�\to \mathrm{\infty }}{lim}\underset{�\to \mathrm{\infty }}{lim}{�}_{�,�}}$ in general.

A sufficient condition of equality is given by the Moore-Osgood theorem, which requires the limit ${\displaystyle \underset{�\to \mathrm{\infty }}{lim}{�}_{�,�}={�}_{�}}$ to be uniform in m.^[10]