Prove IVP with Bolzano-Wierstrass & Heine-Borel

• StarTiger
In summary, the conversation discusses the use of the Bolzano-Wierstrass and Heine-Borel properties to prove a statement about the intermediate value theorem. However, these properties may not be applicable in all cases and may not lead to a valid proof.
StarTiger
Tricky problem. Any tips? Thanks SOO much!

Homework Statement

Let f be continuous on [a,b] and let c be a real number. If for every x in [a,b] f(x) is NOT c, then either f(x) > c for all x in [a,b] OR f(x) < c for all x in [a,b]. Prove this using a) Bolzano-Wierstrass and b) Heine-Borel property.

Homework Equations

B-W property: A set of reals is closed and bounded if and only if every sequence of points chosen fro E has a subsequence that converges to a point in E.
H-B property: A subset of the reals has the HB property if and only if A is both closed and bounded.

(Note the function given fits HB and BW propertis by definition).

The Attempt at a Solution

Some hints:

For WB: suppose false. Explain how there exist sequences {x_n} and {y_n} such that f(x_n) > c, f(y_n) < c and |x_n - y_n| < 1/n
For HB: Suppose false and xplain why there should exist at each point x in [a,b] an open interval I_x centered so that either f(t)>c for all t in intersection of I_x and [a,b] or else f(t)<c for all t in the intersection of I_x in [a,b]

This seems like a strange question. The Heine-Borel and Bolzano-Weierstrass theorems each state that a closed and bounded subset of $$\mathbb{R}$$ is compact (each using a different characterization of compactness). However, the intermediate value theorem for closed intervals in $$\mathbb{R}$$ is not a consequence of compactness, but of connectedness.

In your question, one might replace $$[a,b]$$ by $$[0,1] \cup [2,3]$$, a compact but disconnected set. This set possesses the Heine-Borel and Bolzano-Weierstrass properties, but the intermediate value theorem is obviously false for it (consider any function which takes one constant value on $$[0,1]$$ and another on $$[2,3]$$).

While the fact you are asked to prove is true, the theorems you are asked to use to prove it seem totally inappropriate to the task.

What is the Bolzano-Wierstrass Theorem?

The Bolzano-Wierstrass Theorem states that if a sequence of real numbers is bounded and infinite, then it has at least one accumulation point. This means that the sequence has a subsequence that converges to that accumulation point.

What is the Heine-Borel Theorem?

The Heine-Borel Theorem states that a subset of real numbers is compact if and only if it is closed and bounded. This means that every open cover of the subset has a finite subcover.

How do the Bolzano-Wierstrass and Heine-Borel Theorems relate to IVP?

The Bolzano-Wierstrass Theorem and Heine-Borel Theorem are often used in the proof of the Initial Value Problem (IVP) in calculus. They help to show that a function that satisfies certain conditions at an initial point must also satisfy those conditions at every point in its domain.

Can the Bolzano-Wierstrass and Heine-Borel Theorems be used interchangeably?

No, the Bolzano-Wierstrass and Heine-Borel Theorems are not interchangeable. While both theorems deal with the concept of compactness, they have different conditions and implications. The Bolzano-Wierstrass Theorem deals with sequences, while the Heine-Borel Theorem deals with sets.

How do the Bolzano-Wierstrass and Heine-Borel Theorems apply to real-world phenomena?

The Bolzano-Wierstrass and Heine-Borel Theorems have applications in various areas of science and engineering, including physics, economics, and computer science. They can be used to prove the existence of solutions to differential equations, optimization problems, and more.

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