Table of Contents
How do you prove the Bolzano-Weierstrass Theorem?
proof. Let (sn) be a bounded, nondecreasing sequence. Let S denote the set {sn:n∈N} { s n : n ∈ ℕ } . Then let b=supS (the supremum of S .)…proof of Bolzano-Weierstrass Theorem.
Title | proof of Bolzano-Weierstrass Theorem |
---|---|
Classification | msc 40A05 |
Classification | msc 26A06 |
What is the statement of Bolzano-Weierstrass Theorem?
The theorem states that each bounded sequence in Rn has a convergent subsequence. An equivalent formulation is that a subset of Rn is sequentially compact if and only if it is closed and bounded. The theorem is sometimes called the sequential compactness theorem.
Is the converse of Bolzano-Weierstrass Theorem true?
Bolzano-Weierstrass theorem states that every bounded sequence has a limit point. But, the converse is not true. That is, there are some unbounded sequences which have a limit point.
Why is Bolzano-Weierstrass important?
The Bolzano-Weierstrass theorem is an important and powerful result related to the so-called compactness of intervals , in the real numbers, and you may well see it discussed further in a course on metric spaces or topological spaces.
Does bounded imply convergence?
The corresponding result for bounded below and decreasing follows as a simple corollary. Theorem. If (a_n) is increasing and bounded above, then (a_n) is convergent.
Why is the the Weierstrass approximation theorem important?
The Stone–Weierstrass theorem is a vital result in the study of the algebra of continuous functions on a compact Hausdorff space. A different generalization of Weierstrass’ original theorem is Mergelyan’s theorem, which generalizes it to functions defined on certain subsets of the complex plane.
How do you determine if a sequence is monotonic and bounded?
If {an} is an increasing sequence or {an} is a decreasing sequence we call it monotonic. If there exists a number m such that m≤an m ≤ a n for every n we say the sequence is bounded below.
Can an increasing sequence converge?
Informally, the theorems state that if a sequence is increasing and bounded above by a supremum, then the sequence will converge to the supremum; in the same way, if a sequence is decreasing and is bounded below by an infimum, it will converge to the infimum.
Do all monotone sequences converge?
Not all bounded sequences, like (−1)n, converge, but if we knew the bounded sequence was monotone, then this would change. if an ≥ an+1 for all n ∈ N. A sequence is monotone if it is either increasing or decreasing. and bounded, then it converges.