# Sequence

A **sequence** is an ordered list of terms. Sequences may be either finite or infinite.

## Definition

A sequence of real numbers is simply a function . For instance, the function defined on corresponds to the sequence .

## Convergence

Intuitively, a sequence **converges** if its terms approach a particular number.

Formally, a sequence of reals converges to if and only if for all positive reals , there exists a positive integer such that for all integers , we have . If converges to , is called the limit of and is written . The statement that converges to can be written as .

A classic example of convergence would be to show that as .

**Claim**: .

*Proof*: Let be arbitrary and choose . Then for we see that

which proves that , so as

## Monotone Sequences

Many significant sequences have their terms continually increasing, such as , or continually decreasing, such as . This motivates the following definitions:

A sequence of reals is said to be

**increasing**if for all and**strictly increasing**if for all ,**decreasing**if for all and**strictly decreasing**if for all ,**monotone**if it is either decreasing or increasing.

## Resources

## See Also

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