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

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Lesson Parent:

The Logical Framework For Definite Integrals

Problem:

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

Answer:

When $\forall\epsilon\gt 0(\exists\delta\gt 0(\forall P(||P\,||\lt\delta\Rightarrow U(P)-L(P)\lt\epsilon)))$ is true, then $\phi=\Phi$ .

Solution:

Let's assume that we have a partition $P\,$ such that <Sign in to see all the formulas>. Therefore, <Sign in to see all the formulas>. In general we know that

<Sign in to see all the formulas>

Therefore,

<Sign in to see all the formulas>

and that

<Sign in to see all the formulas>

Putting these together we get

<Sign in to see all the formulas>

Because <Sign in to see all the formulas> this implies that <Sign in to see all the formulas>. Otherwise if <Sign in to see all the formulas> then there could be an <Sign in to see all the formulas> that is small enough that <Sign in to see all the formulas>. But we also know that <Sign in to see all the formulas>. The only way for <Sign in to see all the formulas> and <Sign in to see all the formulas> to be true is if <Sign in to see all the formulas>.

This completes the proof.

Finally, it is important to know that this relationship is even stronger and we can show that

Prove, for a bounded function on [a,b][a,b], that if ∀ϵ>0(∃δ>0(∀P(||P||<δ⇒U(P)−L(P)<ϵ))) \forall\epsilon\gt 0(\exists\delta\gt 0(\forall P(||P\,||\lt\delta\Rightarrow U(P)-L(P)\lt\epsilon))) is true, then ϕ=Φ\phi=\Phi.

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SAMPLE SOLUTION

<Sign in to see all the formulas>

<Sign in to see all the formulas>

<Sign in to see all the formulas>

<Sign in to see all the formulas>

<Sign in to see all the formulas>

GAIN AN ADVANTAGE

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