How do you use the Weierstrass approximation 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….External links.

Can continuous function be approximated by polynomials?

. In words, any continuous function on a closed and bounded interval can be uniformly approximated on that interval by polynomials to any degree of accuracy.

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.

Is 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.

How do you prove the extreme value theorem?

Proof of the Extreme Value Theorem

1. If a function f is continuous on [a,b], then it attains its maximum and minimum values on [a,b].
2. We prove the case that f attains its maximum value on [a,b].
3. Since f is continuous on [a,b], we know it must be bounded on [a,b] by the Boundedness Theorem.

Are polynomials dense in L2?

Polynomials are dense in weighted L2 space.

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.

How do you prove Heine Borel Theorem?

Proof

1. If a set is compact, then it must be closed.
2. If a set is compact, then it is bounded.
3. A closed subset of a compact set is compact.
4. If a set is closed and bounded, then it is compact.

Is every monotone sequence convergent?

A sequence is monotone if it is either increasing or decreasing. and bounded, then it converges.