Why is Bolzano-Weierstrass Theorem important?

The Bolzano-Weierstrass Theorem says that no matter how “random” the sequence (xn) may be, as long as it is bounded then some part of it must converge. This is very useful when one has some process which produces a “random” sequence such as what we had in the idea of the alleged proof in Theorem 7.3.

What is Bolzano-Weierstrass Theorem for C?

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 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.

What is meant by convergent subsequence?

A subsequence of a sequence ( ) is an infinite collection of numbers from ( ) in the same order that they appear in that sequence. The main theorem on subsequences is that every subsequence of a convergent sequence ( ) converges to the same limit as ( ) .

Is every Cauchy sequence is convergent?

Theorem. Every real Cauchy sequence is convergent.

What do you mean by Stone Weierstrass Theorem?

In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval [a, b] can be uniformly approximated as closely as desired by a polynomial function. His result is known as the Stone–Weierstrass theorem.

Is every sequence has a convergent subsequence?

The nicest thing about these subsequences is a result attributed to the Czech mathematician and philosopher Bernard Bolzano (1781 to 1848) and the German mathematician Karl Weierstrass (1815 to 1897). Every bounded sequence has a convergent subsequence.

Can a Cauchy sequence diverge?

Each Cauchy sequence is bounded, so it can not happen that ‖xn‖→∞.

Is (- 1 n Cauchy sequence?

1 n − 1 m < 1 n + 1 m . Similarly, it’s clear that −1 n < 1 n ,, so we get that − 1 n − 1 m < 1 n − 1 m . n , 1 m < 1 N < ε 2 . Thus, xn = 1 n is a Cauchy sequence.

Can every function be represented as a polynomial?

∫3√1+x3dx. Unfortunately, not all functions can be expressed as a polynomial. For example, f(x)=sinx cannot be since a polynomial has only finitely many roots and the sine function has infinitely many roots, namely {nπ|n∈Z}.