# Sequence

**sequencessequentialinfinite sequencefinite sequenceinfinite sequencesnon-increasingorderbi-infinite sequencebi-infinite stringordered sequence**

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed.wikipedia

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### String (computer science)

**stringstringscharacter string**

In computing and computer science, finite sequences are sometimes called strings, words or lists, the different names commonly corresponding to different ways to represent them in computer memory; infinite sequences are called streams.

In computer programming, a string is traditionally a sequence of characters, either as a literal constant or as some kind of variable.

### Stream (computing)

**streamstreamsstreaming**

In computing and computer science, finite sequences are sometimes called strings, words or lists, the different names commonly corresponding to different ways to represent them in computer memory; infinite sequences are called streams.

In computer science, a stream is a sequence of data elements made available over time.

### Set (mathematics)

**setsetsmathematical set**

Like a set, it contains members (also called elements, or terms).

Moreover, the order in which the elements of a set are listed is irrelevant (unlike for a sequence or tuple), so is yet again the same set.

### Function (mathematics)

**functionfunctionsmathematical function**

Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers (for infinite sequences) or the set of the first n natural numbers (for a sequence of finite length n).

Such a function is called a sequence, and, in this case the element f_n is called the nth element of sequence.

### Series (mathematics)

**infinite seriesseriespartial sum**

In particular, sequences are the basis for series, which are important in differential equations and analysis.

In modern terminology, any (ordered) infinite sequence of terms (that is, numbers, functions, or anything that can be added) defines a series, which is the operation of adding the a_i one after the other.

### Finite set

**finitefinitelyfinite sets**

Sequences can be finite, as in these examples, or infinite, such as the sequence of all even positive integers (2, 4, 6, ...).

If a set is finite, its elements may be written — in many ways — in a sequence:

### Limit of a sequence

**limitconvergesconvergence**

In fact, every real number can be written as the limit of a sequence of rational numbers, e.g. via its decimal expansion.

In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to".

### Rational number

**rationalrational numbersrationals**

Other examples of sequences include ones made up of rational numbers, real numbers, and complex numbers.

The decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over.

### Exact sequence

**short exact sequencelong exact sequenceexact**

There are many different notions of sequences in mathematics, some of which (e.g., exact sequence) are not covered by the definitions and notations introduced below.

An exact sequence is a sequence, either finite or infinite, of objects and morphisms between them such that the image of one morphism equals the kernel of the next.

### Integer sequence

**sequence of integersconsecutive integersconsecutive numbers**

In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers.

### Bitstream

**bit streambyte streambinary sequence**

A bitstream (or bit stream), also known as binary sequence, is a sequence of bits.

### Pi

**ππ\pi**

For instance, [[pi|]] is the limit of the sequence (3, 3.1, 3.14, 3.141, 3.1415, ...).

An infinite series is the sum of the terms of an infinite sequence.

### Polynomial sequence

**sequencesequence of polynomials**

In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the degree of the corresponding polynomial.

### Subsequence

**subsequencessub-sequencesub-sequences**

A subsequence of a given sequence is a sequence formed from the given sequence by deleting some of the elements without disturbing the relative positions of the remaining elements.

In mathematics, a subsequence is a sequence that can be derived from another sequence by deleting some or no elements without changing the order of the remaining elements.

### Real analysis

**realtheory of functions of a real variablefunction theory**

The notion of a Cauchy sequence is important in the study of sequences in metric spaces, and, in particular, in real analysis.

In mathematics, real analysis is the branch of mathematical analysis that studies the behavior of real numbers, sequences and series of real numbers, and real-valued functions.

### Cauchy sequence

**CauchyCauchy criterionCauchy sequences**

here.

In mathematics, a Cauchy sequence, named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses.

### Compact space

**compactcompact setcompactness**

One such generalization is that a topological space is sequentially compact if every infinite sequence of points sampled from the space has an infinite subsequence that converges to some point of the space.

### Real number

**realrealsreal-valued**

Other examples of sequences include ones made up of rational numbers, real numbers, and complex numbers. A sequence space is a vector space whose elements are infinite sequences of real or complex numbers.

A sequence (x n ) of real numbers is called a Cauchy sequence if for any ε > 0 there exists an integer N (possibly depending on ε) such that the distance is less than ε for all n and m that are both greater than N.

### Indexed family

**familyindicesindex**

The variable n is called an index, and the set of values that it can take is called the index set.

### Topological space

**topologytopological spacestopological structure**

In some contexts, to shorten exposition, the codomain of the sequence is fixed by context, for example by requiring it to be the set R of real numbers, the set C of complex numbers, or a topological space.

A net is a generalisation of the concept of sequence.

### Uncountable set

**uncountableuncountably infiniteuncountably**

A net is a function from a (possibly uncountable) directed set to a topological space.

The diagonalization proof technique can also be used to show that several other sets are uncountable, such as the set of all infinite sequences of natural numbers and the set of all subsets of the set of natural numbers.

### Separable space

**separableseparabilityseparability axiom**

In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.

### Directed set

**directeddirected subsetdirected subsets**

A net is a function from a (possibly uncountable) directed set to a topological space.

In topology, directed sets are used to define nets, which generalize sequences and unite the various notions of limit used in analysis.

### Pattern

**patternsgeometric patternsgeometric pattern**

Listing is most useful for infinite sequences with a pattern that can be easily discerned from the first few elements.

For example, any sequence of numbers that may be modeled by a mathematical function can be considered a pattern.

### Sequence space

**c'' 0 sequence spacesspace of bounded sequences**

A sequence space is a vector space whose elements are infinite sequences of real or complex numbers.

In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers.