Editing Mathematical analysis (section)

== Important concepts ==
=== Metric spaces ===
{{Main|Metric space}}
In [[mathematics]], a metric space is a [[Set (mathematics)|set]] where a notion of [[distance]] (called a [[metric (mathematics)|metric]]) between elements of the set is defined.

Much of analysis happens in some metric space; the most commonly used are the [[real line]], the [[complex plane]], [[Euclidean space]], other [[vector space]]s, and the [[integer]]s. Examples of analysis without a metric include [[measure theory]] (which describes size rather than distance) and [[functional analysis]] (which studies [[topological vector space]]s that need not have any sense of distance).

Formally, a metric space is an [[ordered pair]] <math>(M,d)</math> where <math>M</math> is a set and <math>d</math> is a [[metric (mathematics)|metric]] on <math>M</math>, i.e., a [[Function (mathematics)|function]]

:<math>d \colon M \times M \rightarrow \mathbb{R}</math>

such that for any <math>x, y, z \in M</math>, the following holds:

# <math>d(x,y) \geq 0</math>, with equality [[if and only if]] <math>x = y</math> &nbsp;&nbsp; (''[[identity of indiscernibles]]''),
# <math>d(x,y) = d(y,x)</math> &nbsp;&nbsp; (''symmetry''), and
# <math>d(x,z) \le d(x,y) + d(y,z)</math> &nbsp;&nbsp; (''[[triangle inequality]]'').

By taking the third property and letting <math>z=x</math>, it can be shown that <math>d(x,y) \ge 0</math> &nbsp;&nbsp;&nbsp; (''non-negative'').

=== Sequences and limits ===
{{Main|Sequence}}
{{See also|Limit of a sequence}}
A sequence is an ordered list. Like a [[Set (mathematics)|set]], it contains [[Element (mathematics)|members]] (also called ''elements'', or ''terms''). Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Most precisely, a sequence can be defined as a [[function (mathematics)|function]] whose domain is a [[countable]] [[totally ordered]] set, such as the [[natural numbers]].

One of the most important properties of a sequence is ''convergence''. Informally, a sequence converges if it has a ''limit''. Continuing informally, a ([[#Finite and infinite|singly-infinite]]) sequence has a limit if it approaches some point ''x'', called the limit, as ''n'' becomes very large. That is, for an abstract sequence (''a''<sub>''n''</sub>) (with ''n'' running from 1 to infinity understood) the distance between ''a''<sub>''n''</sub> and ''x'' approaches 0 as ''n'' → ∞, denoted
:<math>\lim_{n\to\infty} a_n = x.</math>