Main Index
Mathematical Analysis
Infinite series and products
Sequences
Subject Index
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Main Index
Mathematical Analysis
Infinite series and products
Sequences
Subject Index
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In a broad sense, a sequence is an ordered list of objects. Its members are called elements or terms. Contrary to the concept of a set, the order of elements matters. Consequently, the exact same elements can appear multiple times at different positions in the sequence, and its each occurrence counts.
In technical terms, a sequence is a function whose domain is a subset of the set of integers, for example the set of positive integers or the set 
where 
is a positive integer. If the range of the function is the set of real or complex numbers, we speak about real or complex sequences, respectively, or generally about number sequences. The cardinality of the domain of the sequence (possibly infinite) is called the length of the sequence.
A sequence is usually indexed by the positive or non-negative integers, both of which form a so-called totally ordered set. The restriction of indices to integers is restrictive in general convergence considerations. One possibility how to avoid such restriction and to unify various notions of limit in general topological spaces is to replace the index set of positive integers by a more general directed set and to use Moore-Smith generalization [1] of the convergence notion. Integer indexed sequences are still useful and can be used to describe the convergence in topological spaces satisfying the first axiom of countability (every point has a countable neighborhood basis). The most common class of such spaces is formed by metric spaces.
Given a sequence
 | (1) |
of elements of a topological space 
satisfying the first axiom of countability, we say that 
is a limit of this sequence and write
if for every neighborhood 
of 
there is a positive integer 
such that 
for all 
.
To ensure the uniqueness of the limit we have to impose further conditions on the underlying topological space. One such common condition is to require that the topological space is a Hausdorff space 1 (or the so-called 
space). This is a topological space 
in which points can be separated by neighborhoods, more precisely, given any two distinct points 
there is a neighborhood 
of 
and a neighborhood 
of 
such that 
and 
are disjoint.
If a sequence has a limit, we say that the sequence is convergent, and that the sequence converges to the limit 
. Otherwise, the sequence is called divergent.
Example (The zipper theorem): If 
and 
both converge to 
, then also the sequence
converges to 
.
In the case of a metric space 
this definition of the limit boils down to the following form: 
is the limit of the sequence (1) if for every 
there is a positive integer 
such that 
for all 
.
Since every metric space is Hausdorff, the limit, if exists, is uniquely determined in metric spaces.
Note that every convergent sequence of metric space is Cauchy 
. Metric space in which every Cauchy sequences has a limit is called complete. For any metric space 
, one can construct a complete metric space, which contains 
(as a dense subspace).
That not every metric space is complete was already known to ancient Greeks (clearly not in this terminology). The sequence of rational numbers defined by 
, and 
for 
, converges to 
which is not a rational number.
In the case of real number sequences we also distinguish the following two subclasses of divergent sequences:
there exists a 
such that 
for all 
, then we write 
, and we say that (1) diverges to 
,
there exists a 
such that 
for all 
, then we write 
, and we say that (1) diverges to 
.In both above case we also say that the sequence possesses an improper limit. If the limit of a number sequence exists (that is it a “finite” number) we say that it has a proper limit.
If a sequence of real numbers does not converge, but also does not diverge to 
or 
that is, it fails to have a proper or improper limit, is said to oscillate.
Example: Every bounded monotonic sequence of real numbers converges and every unbounded sequence diverges.
A null sequence is a sequence that converges to 0.
| 1 | Felix Hausdorff (1868 - 1942) was one of the founders of topology. His original definition of a topological space included the mentioned condition on disjoint neighbourhoods as an axiom. |
| [1] | Moore, E. H., & Smith, H. L. (1922). A general theory of limits. Amer. J. Math., 44, 102-121. |
Cite this web-page as:
Štefan Porubský: Limit of a sequence.