Prove that the set of all lower sums has a least upper bound.

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Lesson Parent: 
The Logical Framework For Definite Integrals
Problem: 

Assume that $f(x)$ is a bounded function on the closed interval $[a,b]$. Prove that the set of all lower sums has a least upper bound.

Answer: 

It is true that the set of all lower sums has a least upper bound. We denote it by $\phi$.

SAMPLE SOLUTION

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Solution: 

In this problem we showed that for any two arbitrary partitions $P$ and $Q$ that <Sign in to see all the formulas>. This means that every upper sum is an upper bound on every lower sum. By the Axiom of Completeness we know that the set of all lower sums has a least upper bound. The proof is now complete.

On this site we call that least upper bound $\phi$. So for a bounded function and any arbitrary partition $P$, we know that <Sign in to see all the formulas>.