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Let f be a bounded function on [a, b]. If P and Q are partitions of [a, b], prove that L( f , P) ≤ U( f , Q).

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.2: Mappings

Problem 25E: 25. Let, where and are non empty, and let and be subsets of . Prove that. Prove that. Prove...

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Question

Let f be a bounded function on [a, b]. If P and Q are partitions of [a, b], prove that
L( f , P) ≤ U( f , Q).

Expert Solution

Step 1

Given $f$ be a bounded function on $[a, b]$ . If $P$ and $Q$ are the partitions of $[a, b]$ . Then

We have to prove that $L (f, P) \leq U (f, Q)$ .

Let $P = \{x_{0}, x_{1}, x_{2}, ......., x_{n}\}$

Where, $a = x_{0} < x_{1} < x_{2} < ....... < x_{n} = b$ .

Since, $f$ is bounded on $[a, b]$ , hence, $f$ is bounded on $[x_{r - 1}, x_{r}]$ where, $r = 1, 2, 3....., n$

Let $M = \sup_{x \in [x_{r - 1}, x_{r}]} f (x) and m = \inf_{x \in [x_{r - 1}, x_{r}]} f (x)$ for $r = 1, 2, 3....., n$ .

Then, $U (f, P) and L (f, P)$ is defined as follows

$U (f, P) = \sum_{r = 1}^{n} M_{r} (x_{r} - x_{r - 1})$ and $L (f, P) = \sum_{r = 1}^{n} m_{r} (x_{r} - x_{r - 1})$ .

We prove that

$\inf_{x \in [x_{r - 1}, x_{r}]} f (x) = - \sup_{x \in [x_{r - 1}, x_{r}]} (- f (x))$

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