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{{short description|On decreasing nested sequences of non-empty compact sets}} ...fter [[Georg Cantor]], about intersections of decreasing nested [[sequence|sequences]] of non-empty compact sets. ...

...ation in [[numerical analysis]] is to [[approximation theory]], where such theorems quantify the difficulty of approximating general functions by functions of ...th>\epsilon_1 \ge \epsilon_2 \ge \ldots</math> be a decreasing sequence of real numbers tending to zero. Then there exists a point ''x'' in ''X'' such tha ...

...The theorem was found by Cauchy in 1821,<ref name="Knopp"/> subsequently a number of related and generalized results were published, in particular by [[Otto ...<math> (a_n)</math> with <math>a_n\to a </math> and a sequence of positive real numbers <math> (p_n)</math> with <math>\frac{1}{p_1+\cdots+p_n} \to 0</math ...

...ade today may influence the [[economic growth]] of a nation for an unknown number of years into the future. In such cases, it is often convenient to model th ...math> is the set of possible outcomes. E.g., it may be the set of positive real numbers, representing the possible annual [[gross domestic product]]. It is ...

...' are canonical patterns of mathematical proofs that [[limit of a sequence|sequences]] or [[limit of a function|functions]] converge to a finite [[limit (mathem ...r the convergence of [[Series (mathematics)|series]], a particular type of sequences corresponding to sums of many terms, are covered in the article on [[conver ...

{{Short description|Counting real roots of a polynomial in an interval}} ...in an interval, and computing the [[parity (mathematics)|parity]] of this number. It was published in 1807 by [[François Budan de Boislaurent]]. ...

...n of supremum.svg|thumb|300px|Every non-empty subset <math>M</math> of the real numbers <math>\mathbb{R}</math> which is bounded from above has a least upp ...d property. For example, the set <math>\mathbb{Q}</math> of all [[rational number]]s with its natural order does ''not'' have the least upper bound property. ...

...function <math>f(x)</math> defined over an interval <math>I</math> in the real line is called an absolutely monotonic function if it has derivatives <math ...real line that is usually considered is the closed-open right half of the real line, that is, the interval <math>[0, \infty)</math>. ...

...] defined on [[c0 space|<math>c_0</math>]], the [[Banach space]] of scalar sequences that converge to zero. ...J.|title=Lower bounds for the Bohnenblust--Hille inequalities: the case of real scalars|journal=Proceedings of the American Mathematical Society|date=2014| ...

...g of decimal digits is actually a [[decimal representation]] for some real number. Depending on the [[construction of the real numbers]] used, completeness may take the form of an [[axiom]] (the '''comp ...

...infinity. Similarly, a '''regularly varying function''' is a function of a real variable whose behaviour at infinity is similar to the behaviour of a [[pow *{{mvar|β}} is a [[real number]], ...

...book |last1=Elsholtz |first1=Christian |last2=Grabner |first2=Peter |title=Number Theory – Diophantine Problems, Uniform Distribution and Applications: Fests The [[circle group]] arises as the special case of the [[real line]] and its subgroup of the [[integers]]. It has [[nilpotency class]] eq ...

...cs)|graph]] whose [[Vertex (graph theory)|vertices]] are a subset of the [[real numbers]]: two different versions were introduced by {{harvtxt|Abraham|Rubi #''X'' can be partitioned into a countable number of subsets such that any pair from the same subset is black. ...

...stence theorem for economical additive bases of every order}}In [[additive number theory]], an area of [[mathematics]], the '''Erdős–Tetali theorem''' is an ...math>r_{\mathcal{B},h}(n)</math> denotes the number of ways that a natural number ''n'' can be expressed as the sum of ''h'' elements of ''B''.<ref>Alternati ...

title=Über eine verallgemeinerung des Borelschen Theorems| ...d that Menger's basis property can be reformulated to the above form using sequences of open covers. ...

...riable, and then one takes the limit as the other variable approaches some number. ...h>n, m \in \mathbf{N}</math>, let <math>a_{n,m} \in \mathbf{R}</math> be a real double sequence. Then there are two forms of iterated limits, namely ...

...e (economics)|preference ordering]] by a real-valued utility function. The theorems were proved by [[Gerard Debreu]] during the 1950s. ...enient to have a single ''utility function'' - a function that maps a real number to each option, such that the utility of option A is larger than that of op ...

..._N</math> and <math>\lambda_1, \dots, \lambda_N</math> be two sequences of real numbers arranged in a non-increasing order. The condition on the two sequences is equivalent to the [[majorization]] condition: <math>\vec d \preceq \vec ...

...first1=Gert |title=Conflicts between generalization, rigor, and intuition: number concepts underlying the development of analysis in 17th-19th century France === In sequences === ...

.../k^2</math> for every positive [[integer]] <math>k</math> and every [[real number]] <math>t.</math> Then the function <math>f</math> defined by the formula ...e-dimensional [[vector space]] <math>X</math> (or <math>\R^{\N}</math>) of real-valued [[sequence]]s. For example, ...