Editing Real analysis (section)

===Sequences===
{{Main|Sequence}}
A '''''sequence''''' is a [[function (mathematics)|function]] whose [[domain of a function|domain]] is a [[countable]], [[totally ordered]] set.<ref name="Sequences intro">{{cite web |title=Sequences intro |url=https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:sequences/x2f8bb11595b61c86:introduction-to-arithmetic-sequences/v/explicit-and-recursive-definitions-of-sequences |website=khanacademy.org}}</ref> The domain is usually taken to be the [[natural number]]s,<ref name="Gaughan">{{cite book|title=Introduction to Analysis |last=Gaughan |first=Edward |publisher=AMS (2009)|isbn=978-0-8218-4787-9|chapter=1.1 Sequences and Convergence|year=2009 }}</ref> although it is occasionally convenient to also consider bidirectional sequences indexed by the set of all integers, including negative indices.

Of interest in real analysis, a '''''real-valued sequence''''', here indexed by the natural numbers, is a map <math>a : \N \to \R : n \mapsto a_n</math>.  Each <math>a(n) = a_n</math> is referred to as a '''''term''''' (or, less commonly, an '''''element''''') of the sequence.  A sequence is rarely denoted explicitly as a function; instead, by convention, it is almost always notated as if it were an ordered ∞-tuple, with individual terms or a general term enclosed in parentheses:<ref>Some authors (e.g., Rudin 1976) use braces instead and write <math>\{a_n\}</math>.  However, this notation conflicts with the usual notation for a [[Set theory|set]], which, in contrast to a sequence, disregards the order and the multiplicity of its elements.</ref>
<math display="block">(a_n) = (a_n)_{n \in \N}=(a_1, a_2, a_3, \dots) .</math>
A sequence that tends to a [[Limit (mathematics)|limit]] (i.e., <math display="inline">\lim_{n \to \infty} a_n</math> exists) is said to be '''convergent'''; otherwise it is '''divergent'''.  (''See the section on limits and convergence for details.'')  A real-valued sequence <math>(a_n)</math> is '''''bounded''''' if there exists <math>M\in\R</math> such that <math>|a_n|<M</math> for all <math>n\in\mathbb{N}</math>.  A real-valued sequence <math>(a_n)</math> is '''''monotonically increasing''''' or '''''decreasing''''' if
<math display="block">a_1 \leq a_2 \leq a_3 \leq \cdots</math>
or
<math display="block">a_1 \geq a_2 \geq a_3 \geq \cdots</math>
holds, respectively.  If either holds, the sequence is said to be '''''monotonic'''''.  The monotonicity is '''''strict''''' if the chained inequalities still hold with <math>\leq</math> or <math>\geq</math> replaced by < or >.

Given a sequence <math>(a_n)</math>, another sequence <math>(b_k)</math> is a '''''subsequence''''' of <math>(a_n)</math> if <math>b_k=a_{n_k}</math> for all positive integers <math>k</math> and <math>(n_k)</math> is a strictly increasing sequence of natural numbers.