Prove that if P and Q are two arbitrary partitions of [a,b], then L(P)≤U(Q).

Primary tabs

Lesson Parent: 
Integration III: Definite Integrals As Upper and Lower Sums
Problem: 

Assume that f(x) is a bounded function on the closed interval [a,b]. Prove that if P and Q are two arbitrary partitions of [a,b], then the upper and lower sums of f(x) are related by L(P)≤U(Q).

Answer: 

It is true that if P and Q are two arbitrary partitions of [a,b], then L(P)≤U(Q).

SAMPLE SOLUTION

All of our problems have fully worked out, logical solutions. We connect every solution to its underlying lesson so you can understand the concepts needed to solve any problem and reinforce your learning by practicing with similar problems. All of the formulas are available to our subscribers, but here is a sample of the problem solution you will see when you log in:



Solution: 

Because f(x) is a bounded function on the closed interval [a,b] we can form the upper and lower sums U(P) and L(P).

First, by the definition of upper and lower sums we know that

<Sign in to see all the formulas>


and

<Sign in to see all the formulas>

Now let's say we combine the partitions P and Q. If P and Q are equal then we can say that <Sign in to see all the formulas> and therefore <Sign in to see all the formulas>. But what about the case where P and Q are not different? When we combine them we get a new partition R. (Mathematicians call the act of combining two sets intersection and write <Sign in to see all the formulas>.) Now since P and Q are not equal, the partition R differs from both P and Q by at least one point.

Recall from this problem that when we add just one point to a partition we get upper and lower sums that are closer together. This means that

<Sign in to see all the formulas>

and

<Sign in to see all the formulas>

From our rules of inequalities we see that

<Sign in to see all the formulas>

and therefore,

<Sign in to see all the formulas>

The proof is now complete.

Note that we could have just as easily shown that <Sign in to see all the formulas> and therefore <Sign in to see all the formulas>. This emphasizes the point that since P and Q are arbitrary that every lower bound is less than or equal to every upper bound. The consequences of this are discussed here.