Prove for a bounded function on $[a,b]$ that $\phi=\Phi$ if and only if for every $\epsilon\gt 0$ there exists a $\delta\gt 0$ such that for all partitions $P$ where $||P\,||\lt\delta$, then $U(P)-L(P)\lt\epsilon$.

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Lesson Parent: 
The Logical Framework For Definite Integrals
Problem: 

Prove for a bounded function on $[a,b]$ that $\phi=\Phi$ if and only if for every $\epsilon\gt 0$ there exists a $\delta\gt 0$ such that for all partitions $P$ where $||P\,||\lt\delta$, then $U(P)-L(P)\lt\epsilon$.

Answer: 

It is true that $\phi=\Phi$ if and only if for every $\epsilon\gt 0$ there exists a $\delta\gt 0$ such that for all partitions $P$ where $||P\,||\lt\delta$, then $U(P)-L(P)\lt\epsilon$.

SAMPLE SOLUTION

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Solution: 

The statement "for every <Sign in to see all the formulas> there exists a <Sign in to see all the formulas> such that for all partitions $P\,$ where <Sign in to see all the formulas>, then <Sign in to see all the formulas>" is equivalent to the following logical statement:

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Now it is important to know that when we see if and only if this means

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In other words, the statement "if and only if" means <Sign in to see all the formulas>. So to prove

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we must show that

The proof is now complete.