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 .

If there is no such , the sequence is said to diverge. 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 . Much stronger theorems in this respect, which require not much more than pointwise convergence, can be obtained if one abandons the Riemann integral and uses the Lebesgue integral instead.

X in the metric space X if the real sequence (d) 0 in R. 0 Difference in the definitions of cauchy sequence in Real Sequence and in Metric space. 0 Trouble understanding negation of definition definition of convergence metric of convergent sequence. As we know, the limit needs to be unique if it exists. A Banach space is a complete normed vector space, i.e. a real or complex vector space on which a norm is defined.


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. Note that latter definition is simply a generalization since number sequences are, of course, -tuple sequences with . In this section, we apply our knowledge about metrics, open and closed sets to limits. We thereby restrict ourselves to the basics of limits.

  • The definitions given earlier for R generalise very naturally.
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  • Every uniformly convergent sequence is locally uniformly convergent.
  • 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.
  • Almost uniform convergence implies almost everywhere convergence and convergence in measure.
  • Note that almost uniform convergence of a sequence does not mean that the sequence converges uniformly almost everywhere as might be inferred from the name.

“Arbitrarily close to the limit ” can also be reflected by corresponding open balls , where the radius needs to be adjusted accordingly. Now, let us try to formalize our heuristic thoughts about a sequence approaching a number arbitrarily close by employing mathematical terms. Plot of 2-tuple sequence for the first 1000 points that seems to head towards a specific point in .

Definition of a convergent sequence in a metric space

Suppose \(x_n \in E\) for infinitely many \(n \in \). Note that represents an open ball centered at the convergence point or limit x. For instance, for we have the following situation, that all points (i.e. an infinite number) smaller than lie within the open ball .

definition of convergence metric

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. Finally, 2-tuple sequence e) converges to the vector . In this post, we study the most popular way to define convergence by a metric. Note that knowledge about metric spaces is a prerequisite.

The equivalence between these two definitions can be seen as a particular case of the Monge-Kantorovich duality. From the two definitions above, it is clear that the total variation distance between probability measures is always between 0 and 2. Let \(\) be a metric space, \(E \subset X\) a closed set and \(\\) a sequence in \(E\) that converges to some \(x \in X\). Let us furthermore connect the concepts of metric spaces and Cauchy sequences.

One-Sided Limit of a Function

At least that’s why I think the limit has to be in the space. Having said that, it is clear that all the rules and principles also apply to this type of convergence. In particular, this type will be of interest in the context of continuity. Function graph of with singularities at 2Considering the sequence in shows that the actual limit is not contained in . Plot of for b) Let us now consider the sequence that can be denoted by .

definition of convergence metric

A convergent sequence in a metric space has a unique limit. If we already knew the limit in advance, the answer would be trivial. In general, however, the limit is not known and thus the question not easy to answer. It turns out that the Cauchy-property of a sequence is not only necessary but also sufficient. That is, every convergent Cauchy sequence is convergent and every convergent sequence is a Cauchy sequence . Let us re-consider Example 3.1, where the sequence a) apparently converges towards .

Convergence of measures

In an Euclidean space every Cauchy sequence is convergent. 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. We can illustrate that on the real line using balls (i.e. open intervals) as follows.

definition of convergence metric

As mentioned before, this concept is closely related to continuity. Let denote the standard metric space on the real line with and . If the sequence of pushforward measures ∗ converges weakly to X∗ in the sense of weak convergence of measures on X, as defined above.

Almost uniform convergence

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.

Limits of a Sequence

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).

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 \(x_n \in U\). A metric space is called complete if every Cauchy sequence of points in has a limit that is also in .

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. 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 . 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.

Convergence and topology

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. 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.

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. 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. Latter concept is very closely related to continuity at a point. 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.

The range of the function only comprises two real figures . Share a link to this question via email, Twitter, or Facebook. The proofs of the following propositions are left as exercises. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. 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.

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. 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. 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. We first define uniform convergence for real-valued functions, although the concept is readily generalized to functions mapping to metric spaces and, more generally, uniform spaces . Note that the proof is almost identical to the proof of the same fact for sequences of real numbers. In fact many results we know for sequences of real numbers can be proved in the more general settings of metric spaces.