What does Bolzano-Weierstrass theorem state?
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.
How do you prove the Bolzano-Weierstrass Theorem?
Then snk+1
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.
What do you mean by Stone Weierstrass Theorem?
From Wikipedia, the free encyclopedia. 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.
Why is the Bolzano Weierstrass theorem important?
A very important theorem about subsequences was introduced by Bernhard Bolzano and, later, independently proven by Karl Weierstrass. Basically, this theorem says that any bounded sequence of real numbers has a convergent subsequence.
What are bounded sequences?
A sequence is bounded if it is bounded above and below, that is to say, if there is a number, k, less than or equal to all the terms of sequence and another number, K’, greater than or equal to all the terms of the sequence. Therefore, all the terms in the sequence are between k and K’.
Does bounded imply convergence?
Every bounded sequence is NOT necessarily convergent.
Is the sequence n bounded?
If U is an upper bound then so is any number greater than U. If L is a lower bound then so is any number less than L. Bounds are not unique. The sequence (n) is bounded below but is not bounded above because for each value C there exists a number n such that n>C.