Bálsamo de limón

In analyzes mathematical, a continuation of Cauchy is a continuation realities, complexes, points of a metric Espace, or of a uniform topological Espace whose terms approach starting from a certain row. These continuations are those likely to converge. They are in the center of the definition of Complétude. The series Cauchy bear the name of the Mathématicien French Augustin Louis Cauchy.

There exists an equivalent concept for the filter S: the filters of Cauchy .

Real or complex continuation of Cauchy

The difference of the consecutive terms of the continuation $(\backslash \; ln\; (N))$ tends towards 0. One can specify the Speed of convergence:

$\backslash \; ln\; (n+1)\; -\; \backslash \; ln\; (N)\; =\; \backslash \; ln\; \backslash \; left\; (1+\; \backslash \; frac\; \{1\}\; \{N\}\; \backslash \; right)\; =\; \backslash \; frac\; \{1\}\; \{N\}\; +O\; (1/n^2)$. However, $\backslash \; ln\; (2n)\; -\; \backslash \; ln\; (N)\; =\; \backslash \; ln\; (2)$ does not converge towards 0 when N tends towards the infinite one. This observation measures a defect of nonconvergence of the continuation $\backslash \; ln\; (N)$ and led to state a criterion of convergence, the criterion of Cauchy.

A succession of realities or complexes is known as of Cauchy when the terms of the continuation uniformly approach the ones the others in the infinite one with the direction where:

$\backslash \; lim\_\; \{N\; \backslash \; rightarrow\; \backslash \; infty\}\; \backslash \; sup\_\; \{p,\; q>n\}|r\_p-r\_q|=0$. This last condition classically rewrites using universal and existential quantifiers: , ou still: .

Criterion of Cauchy : A continuation $\backslash \; \{r\_n\; \backslash \}$ of realities number or complexes converges if and only if it is a continuation of Cauchy.

Continuation of Cauchy in a metric space

Definition

A continuation $(x\_n)\; \_\; \{N\; \backslash \; in\; \backslash \; mathbb\; NR\}$ in a metric Space $(E,\; d)$ is known as continuation of Cauchy (or of Cauchy ) so for any reality $\backslash \; varepsilon>0$, it exists a Entier naturalness $N$ such as for all entireties $p,\; Q\; \backslash \; geq\; N$, the distance $d\; (x\_p,\; x\_q)$ is lower than $\backslash \; varepsilon$: . The inequalities can be taken indifferently broad or strict. When certain works introduce the concept of continuation of Cauchy only for the continuations of realities, it is exactly the same definition. The distance D is simply to replace by the absolute value of the difference.

Intuitively, the terms of the continuation become increasingly close from/to each other in a certain way which suggests that the continuation must have a limit in space. The convergent continuations are indeed of Cauchy, but nevertheless the reciprocal one is not true in any general information. For example, there exist continuations of rational which are of Cauchy in $\backslash \; mathbb\; Q$ but which does not converge in $\backslash \; mathbb\; Q$.

Exemple: the unit has rational numbers R such as $r^2\; \backslash \; Leq\; 2$ is limited. For entire N , there exists rational a $r\_n$ in has such as $r\_n+1/n$ does not belong to has . The continuation $(r\_n)$ is a succession of rational positive and for p > N , a discussion gives: . Therefore, this continuation is of Cauchy and checks: $r\_n^2\; \backslash \; Leq\; 2\; \backslash \; Leq\; (r\_n+1/n)\; ^2$. If it converges towards rational a L , by passage in extreme cases in the inequalities, one would obtain $l^2=2$. The limit L would be a rational square root of 2, from where a contradiction. The proof of the irrationality of $\backslash \; sqrt\; \{2\}$ does not use the existence of realities. The example given here does not suppose inevitably known the réels.

This is why a metric space in which any continuation of Cauchy converges is known as complete . The whole of the real numbers is complete, and the standard construction of the whole of the real numbers uses the series Cauchy of rational numbers (see the Construction of the real numbers on this subject).

See also: complete Space

Properties

• In a metric space, any convergent continuation is of Cauchy.

Let us suppose that a continuation $x=\; (x\_n)$ of a metric space $(X,\; d)$ converges towards a limit L . Then, for all $\backslash \; epsilon>0$, there exists a sufficiently large entirety $N$ such as for all N > NR one a: . The triangular Inégalité implies that for p , Q > NR , one a: . The continuation X is thus well of Cauchy.

• Any continuation of Cauchy is limited.

That is to say $(x\_n)$ a continuation of Cauchy. Let us apply the definition for $\backslash \; epsilon=1$. There exists a natural entirety $N$ checking for $p,\; Q\; \backslash \; geq\; N$. In particular, for p > NR , one a:. Therefore, starting from the row NR , the terms of the continuation appariennent with a ball of ray 1. Consequently, the continuation X is limited.

• a continuation of Cauchy has with more Valeur of adherence. If it has a value of adherence, then it converges.

A convergent continuation in a metric space has a single value of adherence, namely its limit. The first assertion rises in fact of the second. That is to say X a continuation of Cauchy of $(X,\; d)$ admitting a value of ahérence L . Let us show that X converges towards L . Let us choose an arbitrary reality $\backslash \; epsilon>0$. As X is a continuation of Cauchy, there exists a natural entirety NR such as for all p , and Q > NR , one a: . But L is the limit of a certain continuation extracted from $(x\_n)$, that one notes $(x\_\; \{k\_n\})$ where $(k\_n)$ is a strictly increasing succession of natural entireties. There exists an entirety P such as for all N > P , one a: . One can choose P so that K _P is strictly larger than NR . By triangular Inequality, for N > NR , it comes: . Such an entirety NR could be defined for any reality $\backslash \; epsilon>0$, the continuation X converges towards L .

• the image of a continuation of Cauchy by a application uniformly continues is of Cauchy.

Either F an application uniformly continues of a metric expace $(X,\; d\_X)$ towards $(Y,\; d\_Y)$, and or X a continuation of Cauchy of $(X,\; d\_X)$. Let us fix $\backslash \; epsilon>0$. As F is uniformly continuous, there exists $\backslash \; eta>0$ such as, for all X and x' of X , one a: $d\_X\; (X,\; x\text{'})\; \backslash \; Leq\; \backslash \; eta\; \backslash \; Rightarrow\; d\_Y\; (fx,\; fx\text{'})\; \backslash \; Leq\; \backslash \; epsilon$. As X is of Cauchy, there exists a natural entirety NR such as for all p , Q > NR , one a: . A fortiori, for p , Q > NR , one has by the implication above: . The continuation $(F\; (x\_n))$ is thus itself of Cauchy.

• In the normalized vector spaces, the series Cauchy forms a subspace of the space of the continuations.

A homothety of a normalized vector space is a lipschitzienne application, therefore uniformly continues. The image of a continuation of Cauchy by an application uniformly continues being of Cauchy, if X is a continuation of Cauchy of a vector space normalized E and R is a reality, then R . X is a continuation of Cauchy. In the same way, the sum of two series Cauchy of E is a continuation of Cauchy of E : the vectorial sum defines an application uniformly continues $E\; \backslash \; times\; E\; \backslash \; rightarrow\; E$.

• In a normalized Algebra, a product of series Cauchy is of Cauchy.

Let us consider two series Cauchy X and in a normalized algebra $(has\; there,\; \backslash |.\; \backslash |)$. They are limited (property previously established); then let us note M one raising of the continuations $(x\_n)$ and $(y\_n)$. Let us consider their product $xy$ (produced term in the long term). By definition of the series Cauchy, for $\backslash \; epsilon>0$, there exists an entirety NR such as for all p , Q > NR , one a: and . By triangular inequality, it comes, for p , Q > NR : $\backslash |x\_py\_p-x\_qy\_q\; \backslash |\backslash \; Leq\; \backslash |x\_py\_p-x\_qy\_p\; \backslash |+\; \backslash |x\_qy\_p-x\_qy\_q\; \backslash |\backslash \; Leq\; \backslash |x\_p-x\_q\; \backslash |.\; \backslash |y\_p\; \backslash |+\; \backslash |x\_q\; \backslash |.y\_p\; there\; \_q\; \backslash |\backslash \; Leq\; 2M\; \backslash \; epsilon$. The second inequality comes from under-multiplicativité standard. The continuation $xy$ is thus of Cauchy.

• In a Espace ultrametric (X, d), a continuation $(x\_n)$ is of Cauchy if $d\; (x\_n,\; x\_\; \{n+1\})\; \backslash \; rightarrow\; 0$.

Only the reciprocal direction is not always checked and uses the ultrametric inequality. Thus let us suppose $d\; (x\_n,\; x\_\; \{n+1\})\; \backslash \; rightarrow\; 0$. For $\backslash \; epsilon>0$, there exists a natural entirety NR such as for all N > NR , one a: . By recurrence on K , one shows that for all N > NR , . This property is checked by choice of NR for K =1. Let us suppose established it with the row N , and look at the incrementing. The inequality ultratriangulaire gives: $d\; (x\_n,\; x\_\; \{n+k+1\})\; \backslash \; Leq\; \backslash \; max\; \backslash \; left\; \backslash \; Leq\; \backslash \; epsilon$. The second inequality comes from the application of the assumption of recurrence.

Not standard approaches

In Analysis nonstandard, for a metric space standard $(X,\; d)$, there exists an equivalent but practical definition of the concept of continuation of Cauchy.

• In a metric space standard $(X,\; d)$, a standard continuation X is of Cauchy so for all nonstandard natural entireties N and p , reality $d\; (x\_p,\; x\_q)$ is infinitely small:

$(\backslash \; forall\; p,\; Q\; \backslash \; in\; \backslash \; mathbb\; NR)\; \backslash ;\; (p\; \backslash \; simeq\; \backslash \; infty\; \backslash \; wedge\; Q\; \backslash \; simeq\; \backslash \; infty)\; \backslash \; Rightarrow\; D\; (x\_p,\; x\_q)\; \backslash \; simeq\; 0$.

Indeed, if X is a continuation of Cauchy, then for any reality $\backslash \; epsilon>0$, it exists an entirety $N\; (\backslash \; epsilon)$ such as for all p , Q > NR , one a: . If $\backslash \; epsilon$ is a real standard, the Principe of transfer makes it possible to force on $N\; (\backslash \; epsilon)$ to be a standard entirety because the continuation X is standard. However entire nonstandard naturalness is strictly larger than entire standard naturalness. Therefore, if p and Q is nonstandard entireties, they are larger than all the $N\; (\backslash \; epsilon)$. Of continuation, $d\; (x\_p,\; x\_q)$ is strictly lower than all the real strictly positive standards; it is thus an infinitely small.

Reciprocally, let us suppose that for all nonstandard entireties p and Q , reality $d\; (x\_p,\; x\_q)$ is an infinitely small. Let us fix NR initially a nonstandard entirety. Entire larger than NR is also not standard. That is to say $\backslash \; epsilon>0$ a real standard. Then for p and Q > NR , one a: . In fact, the following assertion:

is checked for all real strictly positive standard $\backslash \; epsilon$. By Principe of transfer, it is checked for all $\backslash \; epsilon>0$, which means exactly that X is of Cauchy.

Continuation of Cauchy in a uniform space

Definitions

In a uniform space, a continuation $(x\_n)$ is known as of Cauchy when for any continuous variation D on X , there exists a natural entirety NR such as for all $p,\; q>N$, one a: .

A family ( X _α) in a uniform Espace X is a family of Cauchy so for any vicinity V it exists a α₀ number such as for all α, β > α₀, the couple ( X _α, X _β) that is to say in V ² .

In practical examples:

• In a topological Group G , a continuation $(g\_n)$ is known as of Cauchy when, for any vicinity V of the neutral element, there exists NR such as for all $p,\; q>N$ one a: $g\_p^\; \{-\; 1\}\; .g\_q\; \backslash \; in\; U$.
• In a topological vector space locally convex E , a series vectors $(u\_n)$ is known as of Cauchy when for any convex vicinity V of 0, there exists a natural entirety NR such as for all p , Q > NR one a: $u\_p-u\_q\; \backslash \; in\; E$.

See too

• Filter of Cauchy

• complete Space
• Cut of Dedekind
• Construction of the real numbers

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