What can be deduced from the Bolzano-Weierstrass Theorem?
The Bolzano–Weierstrass theorem allows one to prove that if the set of allocations is compact and non-empty, then the system has a Pareto-efficient allocation.
What does the Bolzano-weierstrass?
Definition: A set S in a metric space has the Bolzano-Weierstrass Property if every sequence in S has a convergent subsequence — i.e., has a subsequence that converges to a point in S. The B-W Theorem states that closed and bounded (i.e., compact) sets in Rn have the B-W Property.
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.
Is converse of Bolzano-Weierstrass theorem true?
Does bounded imply convergence?
If a sequence an converges, then it is bounded. Note that a sequence being bounded is not a sufficient condition for a sequence to converge. For example, the sequence (−1)n is bounded, but the sequence diverges because the sequence oscillates between 1 and −1 and never approaches a finite number.
What theorem states that any smooth function can be approximated as a polynomial?
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.
What is the purpose of monotone convergence theorem?
In real analysis, the monotone convergence theorem states that if a sequence increases and is bounded above by a supremum, it will converge to the supremum; similarly, if a sequence decreases and is bounded below by an infimum, it will converge to the infimum.
What happens when the convergence is not monotonic?
The sequence in that example was not monotonic but it does converge. Note as well that we can make several variants of this theorem. If {an} is bounded above and increasing then it converges and likewise if {an} is bounded below and decreasing then it converges.
Why do we approximate functions with polynomials?
This is typically done with polynomial or rational (ratio of polynomials) approximations. The objective is to make the approximation as close as possible to the actual function, typically with an accuracy close to that of the underlying computer’s floating point arithmetic.
How do you use the monotone sequence theorem?
Monotone Sequence Theorem: (sn) is increasing and bounded above, then (sn) converges. Intuitively: If (sn) is increasing and has a ceiling, then there’s no way it cannot converge.
What are the conditions of a monotone convergence theorem?
What do you mean by monotone sequence give examples of them?
A sequence is said to be monotone if it is either increasing or decreasing. Example. The sequence n2 : 1, 4, 9, 16, 25, 36, 49, is increasing. The sequence 1/2n : 1/2, 1/4, 1/8, 1/16, 1/32, is decreasing.