Bolzano-weierstrass theorem of metric spaces pdf

We will now look at a rather technical theorem known as the bolzano weierstrass theorem which provides a very important result regarding bounded sequences and convergent subsequences. The bolzanoweierstrass theorem article states that. There is a generalization of bolzanoweierstrass from rn to any finite or infinite dimensional space with a distance or metric. In mathematics, specifically in real analysis, the bolzano weierstrass theorem, named after bernard bolzano and karl weierstrass, is a fundamental result about convergence in a finitedimensional euclidean space rn. A subset a of r n is sequentially compact if and only if it is both closed and bounded. Characterizations of compactness for metric spaces 3 the proof of the main theorem is contained in a sequence of lemmata which we now state. For metric spaces it can be shown that both notions are equivalent, and in this course we will restrict ourselves to the sequential compactness definition given above. Then any sequence xn of points in x has a subsequence. The bolzanoweierstrass theorem is a very important theorem in the realm of analysis. Cauchy criterion, bolzanoweierstrass theorem we have seen one criterion, called monotone criterion, for proving that a sequence converges without knowing its limit. The bolzanoweierstrass theorem is an important and powerful result related to the socalled compactness of intervals, in the real numbers, and you may well see it discussed further in a course on metric spaces or topological spaces. A of open sets is called an open cover of x if every x. Learn about the ttest, the chi square test, the p value and more duration.

Bolzanoweierstrass theorem and sequential compactness a. The theorem states that each bounded sequence in rn has a convergent subsequence. Let be an uncountable regular cardinal with convergent subsequence. Bolzanoweierstrass and heineborel theorems in euclidean spaces. Chapter three deals with continuous functions on metric spaces. Bolzanoweierstrass theorem i a bounded sequence in rp has a convergent sub sequence. Short proof the purpose of this note is to give a short proof of the second version of the bolzanoweierstrass theorem. The proof in wikipedia evidently doesnt go through for an infinitedimensional space, and it seems to me that the theorem ought not to be true in general. A metric space x is said to be sequentially compact if every sequence xn1 n1 of points in x has a convergent subsequence. The bolzanoweierstrass property and compactness we know that not all sequences converge. In the subsequent sections we discuss the proof of the lemmata. In what spaces does the bolzanoweierstrass theorem hold. The proof of the main theorem is contained in a sequence of lemmata which we. The bolzanoweierstrass theorem says that every bounded sequence in rn contains a convergent subsequence.

Note that the completeness of the reals in the form of the monotone convergence theorem is an essential ingredient of the proof. In light of this history, the proof gets its current name. An equivalent formulation is that a subset of rn is sequentially compact if and only if it is closed and bounded. Mathematics department stanford university math 61cm metric spaces we have talked about the notion of convergence in r.

A limit point need not be an element of the set, e. Bolzano weierstrass every bounded sequence has a convergent subsequence. To mention but two applications, the theorem can be used to show that if a. Vice versa let x be a metric space with the bolzanoweierstrass property, i.

Let x be any closed bounded subset of the real line. The ideal course in introductory analysis should help students understand the underpinnings of calculus and to prepare them to dive further into important topics such as measure and probability theory, metric and normed linear spaces, differential geometry, and functional analysis, and to prepare for advanced excursions in differential equations. Mathematics department stanford university math 61cm. A metric space is compact if and only if all sequences have a convergent subsequence. Every bounded sequence in r has a convergent subsequence. Now that we have the bolzanoweierstrass theorem, its time to use it to prove stuff. So i guess you are referring to the theorem being generalized to an arbitrary metric space with bounded and closed being replaced by compact. Mat25 lecture 12 notes university of california, davis. The heineborel theorem does not hold as stated for general metric and topological vector spaces, and this gives rise to the necessity to consider special classes of spaces where this proposition is true. A fundamental tool used in the analysis of the real line is the wellknown bolzanoweierstrass theorem1. Compactness of metric spaces compactness in metric spaces the closed intervals a,b of the real line, and more generally the closed bounded subsets of rn, have some remarkable properties, which i believe you have studied in your course in real analysis.

Completeness and completion compactness in metric spaces. The theorem is sometimes called the sequential compactness theorem. The bolzanoweierstrass theorem applies to spaces other than closed bounded intervals. Eclasses, which we now call metric spaces, and vclasses,15 a metric space with a weak version of the triangle inequality, were less general, but easier to work with.

We saw that bolzanoweierstrass in r implies the compactness of intervals a, b. A set s in a metric space has the bolzanoweierstrass property if every. A metric space is sequentially compact if and only if every in. Every bounded sequence of real numbers has a convergent subsequence. They are called the spaces with the heineborel property. It was first proved by bernhard bolzano but it became well known with the proof by karl weierstrass who did not know about bolzanos proof. For the remainder of this section we discuss a number of examples of cluster point problems of certain spaces.