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Prove for a bounded function on $[a,b]$ that if $\phi=\Phi$ , then 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 if $\phi=\Phi$ , then 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 if $\phi=\Phi$ , then $\forall\epsilon>0(\exists\delta>0(\forall P(||P||<\delta\Rightarrow U(P)-L(P)<\epsilon))).$

Solution:

First, we are going to put this into logical, math symbols so we can easily connect things to what we already know. So 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: