Menu

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 $U(P)-L(P)\lt\epsilon$ .

Primary tabs

Lesson Parent:

The Logical Framework For Definite Integrals

Problem:

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 $U(P)-L(P)\lt\epsilon$ .

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:

<Sign in to see all the formulas>

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

<Sign in to see all the formulas>

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

<Sign in to see all the formulas>

we must show that

<Sign in to see all the formulas>. (We have done so here.)
<Sign in to see all the formulas>. (We have done so here.)

The proof is now complete.

Prove for a bounded function on [a,b][a,b] that ϕ=Φ\phi=\Phi if and only if for every ϵ>0\epsilon\gt 0 there exists a δ>0\delta\gt 0 such that for all partitions PP where ||P||<δ||P\,||\lt\delta, then U(P)−L(P)<ϵU(P)-L(P)\lt\epsilon.

Primary tabs

SAMPLE SOLUTION

<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 $\phi=\Phi$ 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 $U(P)-L(P)\lt\epsilon$ .