# 8 3: Sequences and Convergence Mathematics LibreTexts

In order to define other types of convergence (e.g. point-wise convergence of functions) one needs to extend the following approach based on open sets. Almost uniform convergence implies almost everywhere convergence and convergence in measure. Is in V. In this situation, uniform limit of continuous functions remains continuous. When we take a closure of a set $$A$$, we really throw in precisely those points that are limits of sequences in $$A$$. Again, we will be cheating a little bit and we will use the definite article in front of the word limit before we prove that the limit is unique. The notion of a sequence in a metric space is very similar to a sequence of real numbers.

Plot of for b) Let us now consider the sequence that can be denoted by . The range of the function definition of convergence metric only comprises two real figures . Share a link to this question via email, Twitter, or Facebook.

## Limits of a Sequence

In this post, we study the most popular way to define convergence by a metric. Note that knowledge about metric spaces is a prerequisite. There are several equivalent definitions of weak convergence of a sequence of measures, some of which are more general than others. Just as a convergent sequence in R can be thought of as a sequence of better and better approximtions to a limit, so a sequence of “points” in a metric space can approximate a limit here.

Let $$E \subset X$$ be closed and let $$\$$ be a sequence in $$X$$ converging to $$p \in X$$. In the one-dimensional metric space there are only two ways to approach a certain point on the real line. For instance, the point can be either be approached from the negative or from the positive part of the real line. Sometimes this is stated as the limit is approached “from the left/righ” or “from below/above”.

If you want to get a deeper understanding of converging sequences, the second part (i.e. Level II) of the following video by Mathologer is recommended. Plot of 2-tuple sequence for the first 1000 points that seems to head towards a specific point in .

Let us furthermore connect the concepts of metric spaces and Cauchy sequences. In mathematics and statistics, weak convergence is one of many types of convergence relating to the convergence of measures. It depends on a topology on the underlying space and thus is not a purely measure theoretic notion. That is, for being the metric space the left-sided and the right-sided domains are and , respectively.

## Definition of a convergent sequence in a metric space

Here the supremum is taken over f ranging over the set of all measurable functions from X to [−1, 1]. In the case where X is a Polish space, the total variation metric coincides with the Radon metric. A set is closed when it contains the limits of its convergent sequences. Please note that it also important in what space the process is considered. It might be that a sequence is heading to a number that is not in the range of the sequence (i.e. not part of the considered space). For instance, the sequence Example 3.1 a) converges in to 0, however, fails to converge in the set of all positive real numbers .

If $\$ is a sequence contained in a metric space , then $x_n \rightarrow x$ if and only if $d \rightarrow 0$. Right-sided means that the -value decreases on the real axis and approaches from the right to the limit point . In other words, almost uniform convergence means there are sets of arbitrarily small measure for which the sequence of functions converges uniformly on their complement. A sequence of continuous functions on metric spaces, with the image metric space being complete, is uniformly convergent if and only if it is uniformly Cauchy.

• Function graph of with singularities at 2Considering the sequence in shows that the actual limit is not contained in .
• We use the Balzano-Weierstrass Theorem to show that has an accumulation point , and then we show that converges to .
• In the one-dimensional metric space there are only two ways to approach a certain point on the real line.
• The erroneous claim that the pointwise limit of a sequence of continuous functions is continuous is infamously known as “Cauchy’s wrong theorem”.

This limit process conveys the intuitive idea that can be made arbitrarily close to provided that is sufficiently large. In fact this last result holds for any finite-dimensional space Rn and also holds for such spaces with any of the metrics dp. The situation for infinite-dimensional spaces of sequences or functions is different as we will see in the next section. Your essentially embedding your space in another space where the convergence is standard.

## Convergence in a metric space

Connect and share knowledge within a single location that is structured and easy to search. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Sequences are, basically, countably many (– or higher-dimensional) vectors arranged in an ordered set that may or may not exhibit certain patterns.

Hence, since is infinite there must be an accumulation point according to the Bolzano-Weierstrass Theorem. If an increasing sequence is bounded above, then converges to the supremum of its range. Note that latter definition is simply a generalization since number sequences are, of course, -tuple sequences https://globalcloudteam.com/ with . While he thought it a “remarkable fact” when a series converged in this way, he did not give a formal definition, nor use the property in any of his proofs. “Arbitrarily close to the limit ” can also be reflected by corresponding open balls , where the radius needs to be adjusted accordingly.

Let  be a metric space and $$\$$ a sequence in $$X$$. Then $$\$$ converges to $$x \in X$$ if and only if for every open neighborhood $$U$$ of $$x$$, there exists an $$M \in$$ such that for all $$n \geq M$$ we have $$x_n \in U$$. Note that it is not necessary for a convergent sequence to actually reach its limit. It is only important that the sequence can get arbitrarily close to its limit. In this section, we apply our knowledge about metrics, open and closed sets to limits. Accordingly, a real number sequence is convergent if the absolute amount is getting arbitrarily close to some number , i.e. if there is an integer such that whenever .

The erroneous claim that the pointwise limit of a sequence of continuous functions is continuous is infamously known as “Cauchy’s wrong theorem”. However, Egorov’s theorem does guarantee that on a finite measure space, a sequence of functions that converges almost everywhere also converges almost uniformly on the same set. Every locally uniformly convergent sequence is compactly convergent.

A metric space is called complete if every Cauchy sequence of points in has a limit that is also in . Sequence b) instead is alternating between and and, hence, does not converge. Note that example b) is a bounded sequence that is not convergent. Sequence c) does not have a limit in as it is growing towards and is therefore not bounded. Note that a sequence can be considered as a function with domain . We need to distinguish this from functions that map sequences to corresponding function values.

As before, this implies convergence of integrals against bounded measurable functions, but this time convergence is uniform over all functions bounded by any fixed constant. Property holds for almost all terms of if there is some such that is true for infinitely many of the terms with . While he thought it a “remarkable fact” when a series converged in this way, he did not give a formal definition, nor use the property in any of his proofs.

## Convergence in metric spaces

If the sequence of pushforward measures ∗ converges weakly to X∗ in the sense of weak convergence of measures on X, as defined above. These observations preclude the possibility of uniform convergence. To formalize this requires a careful specification of the set of functions under consideration and how uniform the convergence should be.

Latter concept is very closely related to continuity at a point. Converges uniformly on E then f is integrable on E and the series of integrals of fn is equal to integral of the series of fn. X in the metric space X if the real sequence (d) 0 in R. In the following example, we consider the function and sequences that are interpreted as attributes of this function. If we consider the points of the domain and the function values of the range, we get two sequences that correspond to each other via the function.

## Definition in a hyperreal setting

If we then consider the limit of the restricted functions and , we get an equivalent to the definitions above. In this section it is about the limit of a sequence that is mapped via a function to a corresponding sequence of the range. 0 Difference in the definitions of cauchy sequence in Real Sequence and in Metric space. 0 Trouble understanding negation of definition of convergent sequence. A Banach space is a complete normed vector space, i.e. a real or complex vector space on which a norm is defined. For locally compact spaces local uniform convergence and compact convergence coincide.

We can illustrate that on the real line using balls (i.e. open intervals) as follows. Plot of the sequence e) Consider the 2-tuple sequence in . A) The sequence can be written as and is nothing but a function defined by . As the set of Dirac measures, and its convex hull is dense. The definitions given earlier for R generalise very naturally. In fact the sequence in R2 converges to the point (π, π).

## Total variation convergence of measures

If a sequence converges to a limit , its terms must ultimately become close to its limit and hence close to each other. That is, two arbitrary terms and of a convergent sequence become closer and closer to each other provided that the index of both are sufficiently large. While a sequence in a metric space does not need to converge, if its limit is unique. Notice, that a ‘detour’ via another convergence point would turn out to be the direct path with respect to the metric as .

Completely standard notions of convergence did not exist at the time, and Cauchy handled convergence using infinitesimal methods. When put into the modern language, what Cauchy proved is that a uniformly convergent sequence of continuous functions has a continuous limit. The image above shows a counterexample, and many discontinuous functions could, in fact, be written as a Fourier series of continuous functions. The erroneous claim that the pointwise limit of a sequence of continuous functions is continuous is infamously known as “Cauchy’s wrong theorem”. The uniform limit theorem shows that a stronger form of convergence, uniform convergence, is needed to ensure the preservation of continuity in the limit function.

## Convergence in euclidean space

The definition of convergence implies that if and only if . The convergence of the sequence to 0 takes place in the standard Euclidean metric space . There are several equivalent definitions definition of convergence metric of weak convergence of a sequence of measures, some of which are more general than others.

## Convergence and topology

Convergence actually means that the corresponding sequence gets as close as it is desired without actually reaching its limit. Hence, it might be that the limit of the sequence is not defined at but it has to be defined in a neighborhood of . A sequence that fulfills this requirement is called convergent.