## 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 converse of Bolzano-Weierstrass Theorem?

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 Bolzano theorem?**

PROOF of BOLZANO’s THEOREM: Let S be the set of numbers x within the closed interval from a to b where f(x) < 0. Since S is not empty (it contains a) and S is bounded (it is a subset of [a,b]), the Least Upper Bound axiom asserts the existence of a least upper bound, say c, for S.

**Why is Bolzano weierstrass important?**

History and significance It was actually first proved by Bolzano in 1817 as a lemma in the proof of the intermediate value theorem. Some fifty years later the result was identified as significant in its own right, and proved again by Weierstrass. It has since become an essential theorem of analysis.

## How do you prove Bolzano weierstrass?

Theorem 1 (Bolzano-Weierstrass Theorem, Version 1). Every bounded sequence of real numbers has a convergent subsequence. To mention but two applications, the theorem can be used to show that if [a, b] is a closed, bounded interval and f : [a, b] → R is continous, then f is bounded.

### What is a monotone sequence?

Monotone Sequences. Definition : We say that a sequence (xn) is increasing if xn ≤ xn+1 for all n and strictly increasing if xn < xn+1 for all n. Similarly, we define decreasing and strictly decreasing sequences. Sequences which are either increasing or decreasing are called monotone.

**How do you prove the intermediate value theorem?**

Proof of the Intermediate Value Theorem

- If f(x) is continuous on [a,b] and k is strictly between f(a) and f(b), then there exists some c in (a,b) where f(c)=k.
- Without loss of generality, let us assume that k is between f(a) and f(b) in the following way: f(a)

**What is monotonic and bounded sequence?**

In this section, we will be talking about monotonic and bounded sequences. We will learn that monotonic sequences are sequences which constantly increase or constantly decrease. We also learn that a sequence is bounded above if the sequence has a maximum value, and is bounded below if the sequence has a minimum value.

## What is the monotonic 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.

### How do you prove boundedness?

To show that f attains its bounds, take M to be the least upper bound of the set X = { f (x) | x ∈ [a, b] }. We need to find a point β ∈ [a, b] with f (β) = M . To do this we construct a sequence in the following way: For each n ∈ N, let xn be a point for which | M – f (xn) | < 1/n.

**Is Weierstrass function differentiable?**

In mathematics, the Weierstrass function is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is an example of a fractal curve.