Editing Real analysis (section)

===Limits and convergence===
{{Main|Limit (mathematics)}}
Roughly speaking, a '''limit''' is the value that a [[function (mathematics)|function]] or a [[sequence]] "approaches" as the input or index approaches some value.<ref>{{cite book|last=Stewart|first=James|author-link=James Stewart (mathematician)|title=Calculus: Early Transcendentals|publisher=[[Brooks/Cole]]|edition=6th|year=2008|isbn=978-0-495-01166-8|url=https://archive.org/details/calculusearlytra00stew_1}}</ref> (This value can include the symbols <math>\pm\infty</math> when addressing the behavior of a function or sequence as the variable increases or decreases without bound.)  The idea of a limit is fundamental to [[calculus]] (and [[mathematical analysis]] in general) and its formal definition is used in turn to define notions like [[continuous function|continuity]], [[derivative]]s, and [[integral]]s.  (In fact, the study of limiting behavior has been used as a characteristic that distinguishes calculus and mathematical analysis from other branches of mathematics.)

The concept of limit was informally introduced for functions by [[Isaac Newton|Newton]] and [[Gottfried Wilhelm Leibniz|Leibniz]], at the end of the 17th century, for building [[infinitesimal calculus]]. For sequences, the concept was introduced by [[Augustin-Louis Cauchy|Cauchy]], and made rigorous, at the end of the 19th century by [[Bernard Bolzano|Bolzano]] and [[Karl Weierstrass|Weierstrass]], who gave the modern [[ε-δ definition]], which follows.

'''Definition.''' Let <math>f</math> be a real-valued function defined on {{nowrap|<math>E\subset\mathbb{R}</math>.}}  We say that '''''<math>f(x)</math> tends to <math>L</math> as <math>x</math> approaches <math>x_0</math>''''', or that '''''the limit of <math>f(x)</math> as <math>x</math> approaches <math>x_0</math> is <math>L</math>''''' if, for any <math>\varepsilon>0</math>, there exists <math>\delta>0</math> such that for all <math>x\in E</math>, <math>0 < |x - x_0| < \delta</math> implies that <math>|f(x) - L| < \varepsilon</math>.  We write this symbolically as
<math display="block">f(x)\to L\ \ \text{as}\ \ x\to x_0 ,</math> or as <math display="block">\lim_{x\to x_0} f(x) = L .</math>
Intuitively, this definition can be thought of in the following way:  We say that <math>f(x)\to L</math> as <math>x\to x_0</math>, when, given any positive number <math>\varepsilon</math>, no matter how small, we can always find a <math>\delta</math>, such that we can guarantee that <math>f(x)</math> and <math>L</math> are less than <math>\varepsilon</math> apart, as long as <math>x</math> (in the domain of <math>f</math>) is a real number that is less than <math>\delta</math> away from <math>x_0</math> but distinct from <math>x_0</math>.  The purpose of the last stipulation, which corresponds to the condition <math>0<|x-x_0|</math> in the definition, is to ensure that <math display="inline">\lim_{x \to x_0} f(x)=L</math> does not imply anything about the value of <math>f(x_0)</math> itself.  Actually, <math>x_0</math> does not even need to be in the domain of <math>f</math> in order for <math display="inline">\lim_{x \to x_0} f(x)</math> to exist.

In a slightly different but related context, the concept of a limit applies to the behavior of a sequence <math>(a_n)</math> when <math>n</math> becomes large.

'''Definition.'''  Let <math>(a_n)</math> be a real-valued sequence.  We say that <math>(a_n)</math> '''''converges to''''' <math>a</math> if, for any <math>\varepsilon > 0</math>, there exists a natural number <math>N</math> such that <math>n\geq N</math> implies that <math>|a-a_n| < \varepsilon</math>.  We write this symbolically as
<math display="block">a_n \to a\ \ \text{as}\ \ n \to \infty ,</math>or as<math display="block">\lim_{n \to \infty} a_n = a ;</math>
if <math>(a_n)</math> fails to converge, we say that <math>(a_n)</math> '''''diverges'''''.

Generalizing to a real-valued function of a real variable, a slight modification of this definition (replacement of sequence <math>(a_n)</math> and term <math>a_n</math> by function <math>f</math> and value <math>f(x)</math> and natural numbers <math>N</math> and <math>n</math> by real numbers <math>M</math> and <math>x</math>, respectively) yields the definition of the '''''limit of <math>f(x)</math> as <math>x</math> increases without bound''''', notated <math display="inline">\lim_{x \to \infty} f(x)</math>.  Reversing the inequality <math>x\geq M</math> to <math>x \leq M</math> gives the corresponding definition of the limit of <math>f(x)</math> as <math>x</math> ''decreases'' ''without bound'', {{nowrap|<math display="inline">\lim_{x \to -\infty} f(x)</math>.}}

Sometimes, it is useful to conclude that a sequence converges, even though the value to which it converges is unknown or irrelevant.  In these cases, the concept of a [[Cauchy sequence]] is useful.

'''Definition.'''  Let <math>(a_n)</math> be a real-valued sequence.  We say that <math>(a_n)</math> is a '''''Cauchy sequence''''' if, for any <math>\varepsilon > 0</math>, there exists a natural number <math>N</math> such that <math>m,n\geq N</math> implies that <math>|a_m-a_n| < \varepsilon</math>.

It can be shown that a real-valued sequence is Cauchy [[if and only if]] it is convergent.  This property of the real numbers is expressed by saying that the real numbers endowed with the standard metric, <math>(\R, |\cdot|)</math>, is a '''''[[complete metric space]]'''''.  In a general metric space, however, a Cauchy sequence need not converge.

In addition, for real-valued sequences that are monotonic, it can be shown that the sequence is bounded if and only if it is convergent.

==== Uniform and pointwise convergence for sequences of functions ====
{{Main|Uniform convergence}}
In addition to sequences of numbers, one may also speak of ''sequences of functions'' ''on'' <math>E\subset \mathbb{R}</math>, that is, infinite, ordered families of functions <math>f_n:E\to\mathbb{R}</math>, denoted <math>(f_n)_{n=1}^\infty</math>, and their convergence properties.  However, in the case of sequences of functions, there are two kinds of convergence, known as ''pointwise convergence'' and ''uniform convergence'', that need to be distinguished.

Roughly speaking, pointwise convergence of functions <math>f_n</math> to a limiting function <math>f:E\to\mathbb{R}</math>, denoted <math>f_n \rightarrow f</math>, simply means that given any <math>x\in E</math>, <math>f_n(x)\to f(x)</math> as <math>n\to\infty</math>.  In contrast, uniform convergence is a stronger type of convergence, in the sense that a uniformly convergent sequence of functions also converges pointwise, but not conversely.  [[Uniform convergence]] requires members of the family of functions, <math>f_n</math>, to fall within some error <math>\varepsilon > 0</math> of <math>f</math> for ''every value of <math>x\in E</math>'', whenever <math>n\geq N</math>, for some integer <math>N</math>.  For a family of functions to uniformly converge, sometimes denoted <math>f_n\rightrightarrows f</math>, such a value of <math>N</math> must exist for any <math>\varepsilon>0</math> given, no matter how small.  Intuitively, we can visualize this situation by imagining that, for a large enough <math>N</math>, the functions <math>f_N, f_{N+1}, f_{N+2},\ldots</math> are all confined within a 'tube' of width <math>2\varepsilon</math> about <math>f</math> (that is, between <math>f - \varepsilon</math> and <math>f+\varepsilon</math>) ''for every value in their domain'' <math>E</math>.

The distinction between pointwise and uniform convergence is important when exchanging the order of two limiting operations (e.g., taking a limit, a derivative, or integral) is desired: in order for the exchange to be well-behaved, many theorems of real analysis call for uniform convergence.  For example, a sequence of continuous functions (see [[Real analysis#Continuity|below]]) is guaranteed to converge to a continuous limiting function if the convergence is uniform, while the limiting function may not be continuous if convergence is only pointwise.  [[Karl Weierstrass]] is generally credited for clearly defining the concept of uniform convergence and fully investigating its implications.