Metric spaces and sequences proof pdf

The metric is a function that defines a concept of distance between any two members of the set, which are usually called points. In the next section we shall see how we can define convergence and continuity for sequences and functions in metric spaces. Metric spaces a metric space is a set x that has a notion of the distance dx,y between every pair of points x,y. Suppose x n is a convergent sequence which converges to two di. The function dis called the metric, it is also called the distance function. Often, if the metric dis clear from context, we will simply denote the metric space x. This chapter will introduce the reader to the concept of metrics a class of functions which is regarded as generalization of the notion of distance and metric spaces. Banach spaces these notes provide an introduction to banach spaces, which are complete normed vector spaces. Xis closed and x n is a cauchy sequence in f, then x n. Complete metric spaces notions such as convergent sequence and cauchy sequence make sense for any metric space.

A point x is called limit point of the sequence x nn 2 n 2 m n if there is a subsequence n jj2 n of n n 2 n such that x n j. The proof that this is a metric follows the same pattern as the case n 2 given in the previous example. Then a is complete if and only if a is closed in x. Since is a complete space, the sequence has a limit. Namely, take any set x whatsoever, and define a metric 0 on x by the rule 0, 0, 1. Chapter i normed vector spaces, banach spaces and metric spaces 1 normed vector spaces and banach spaces in the following let xbe a linear space vector space over the eld f 2fr. As we said, the standard example of a metric space is rn, and r, r2, and r3 in particular. We do not develop their theory in detail, and we leave the veri. Recall that in a euclidean space the scalar product is defined by eq. A metric space x is sequentially compact if every sequence of points in x has a convergent subsequence converging to a point in x.

Thus we conclude the discrete metric space is complete. Ais a family of sets in cindexed by some index set a,then a o c. A subset of a complete metric space is itself a complete metric space if and only if it is closed. A metric space is called complete if every cauchy sequence converges to a limit. For the purposes of these notes, all vector spaces are assumed to be over the real numbers. The definitions given earlier for r generalise very naturally. Introduction let x be an arbitrary set, which could consist of vectors in rn, functions, sequences, matrices, etc. It helps to have a unifying framework for discussing both random variables and stochastic processes, as well as their convergence, and such a framework is provided by metric spaces. I must check that the three metric properties hold. In mathematics, a metric space is a set together with a metric on the set. C mappings, and derive a fixed point theorem fpt on the mappings in the space.

Indeed, if a sequence is convergent, then it is cauchy it cant be not cauchy, you have just proved that. A metric space is a set xtogether with a metric don it, and we will use the notation x. In a complete metric space, the following variant of cantors intersection theorem holds. Metric spaces constitute an important class of topological spaces. The usual proofs either use the lebesgue number of an open cover or. In other words, no sequence may converge to two different limits. Let x,d be a given metric space and let xn be a sequence of points of x. Moreover, we introduce two kinds of notions of generalized fuzzy.

Heres a proof that the sum of two convergent sequences in a normed vector. Introduction when we consider properties of a reasonable function, probably the. Given 0 n n such that n n dx n, x metric space x if the real sequence dx n, x 0 in r. A sequence in a set xa sequence of elements of x is a function s. In the usual notation for functions the value of the function x at the integer n is written x n, but whe we discuss sequences we will always write xn instead of x n. Metric spaces, topological spaces, and compactness proposition a. Pdf we studied the relation between crisp sequence and fuzzy sequence in metric space, and we gave a prove for the theorem that if a family. We give the definition of cauchy sequence in metric spaces, prove that every cauchy sequence is convergent, and motivate discussion with example.

Sequences and closed sets we can characterize closedness also using sequences. A space where all cauchy sequences are convergent, is called a complete space. Characterizations of compactness for metric spaces 3 the proof of the main theorem is contained in a sequence of lemmata which we now state. Metric spaces proofs covered in class tcd maths home. The limit of a sequence in a metric space is unique. For any sequence xn we can consider the set of values it attains. In the subsequent sections we discuss the proof of the lemmata. If a complete metric space has a norm defined by an inner product such as in a euclidean space, it is called a hilbert space. Convergence of a sequence in a metric space part 1. The fundamental concepts of metric spaces will be established and explored with some instructive examples.

A metric space is sequentially compact if and only if every in. Since a sequence in a metric space x, d is a function from n into x, the definition. Tripled fuzzy metric spaces and fixed point theorem. For example, if dis the discrete metric, a sequence converges if and only if all the terms from some. Then we present some related examples, topological properties, convergence of sequences, cauchy sequence cs and completeness of the tfms. A metric space is a set x that has a notion of the distance dx,y between every pair of points x,y. Xthe number dx,y gives us the distance between them. Schep in this note we shall present a proof that in a metric space x. Definition the sequence x n in a metric space is convergent to x x if. Many, although not all, of the metric spaces used in economics are derived from normed vector spaces.

Let be a cauchy sequence in the sequence of real numbers is a cauchy sequence check it. So in discrete metric space, every cauchy sequence is constant sequence and that way every cauchy sequence is convergent sequence. A metric space in which every cauchy sequence is a convergent sequence is a complete space. Informally, 3 and 4 say, respectively, that cis closed under. We studied the relation between crisp sequence and fuzzy sequence in metric space, and we gave a prove for the.

Contraction mappings and the contraction mapping principle will also be explored. A metric space is a set in which we can talk of the distance between any two of its elements. Metric spaces and the contraction mapping principle luke peeler abstract. Every convergent sequence in a metric space is a cauchy sequence. Suppose we take the same metric and sequence as before, but now x 0,1. A metric space is a pair x, d, where x is a set and d is a metric on x. Convergence of a sequence in a metric space part 1 youtube. The definition below imposes certain natural conditions on the distance between the points. Suppose that x is a complete metric space, and is a sequence of nonempty closed nested subsets of x whose diameters tend to zero.

When we prove theorems about these concepts, they automatically hold in all metric spaces, saving us the labor of having to prove them over and over again each time we introduce a new class of spaces. Y continuous if and only if, for every cauchy sequence fx igin x converging to x2x, lim i fx i f lim i x i proof. We introduce metric spaces and give some examples in. The proof of the triangle inequality follows the same form as in that case. Pdf further proof on fuzzy sequences on metric spaces.