# Heine-Borel Theorem

Not what you're looking for?

Let f:[a,b]--> R be continuous. Prove that the set f([a,b]) has both a least upper bound M and a greatest lower bound m and that there are points u,v in [a,b] such that f(u)=M, f(v)=m.

##### Purchase this Solution

##### Solution Summary

Extreme Value Theorem from calculus is infused in this solution.

##### Solution Preview

Let be continuous. Prove that the set has both a least upper bound M and a greatest lower bound m and that there are points u, v, in such that and

Proof: Note that this is the Extreme Value Theorem from calculus. To prove it, we can make use of two theorems from topology.

Theorem 1. Heine-Borel Theorem. A set is compact if and only if E is closed and bounded.

Theorem 2. Let ...

##### Purchase this Solution

##### Free BrainMass Quizzes

##### Graphs and Functions

This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.

##### Geometry - Real Life Application Problems

Understanding of how geometry applies to in real-world contexts

##### Solving quadratic inequalities

This quiz test you on how well you are familiar with solving quadratic inequalities.

##### Know Your Linear Equations

Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.

##### Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.