The goal of this question is to prove the following result from the definition of integral: Theorem 1. Let a 0, there exists a partition P of [a, b] such that Up(f) – Lp(ƒ) < €. Hint: you may need to use the fact about real numbers: Let x = R. IF Ve > 0, x < € THEN x ≤ 0. (b) Prove that if f is integrable on [a, b] then so is f².
The goal of this question is to prove the following result from the definition of integral: Theorem 1. Let a 0, there exists a partition P of [a, b] such that Up(f) – Lp(ƒ) < €. Hint: you may need to use the fact about real numbers: Let x = R. IF Ve > 0, x < € THEN x ≤ 0. (b) Prove that if f is integrable on [a, b] then so is f².
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![The goal of this question is to prove the following result from the definition
of integral:
Theorem 1. Let a <b. Let f and g be bounded functions on [a, b].
If ƒ and g are integrable on [a, b]
Then fg is integrable on [a, b].
We will break the proof into pieces and guide you through them. In
mathematical terms, you will be proving a few "lemmas" before you prove
Theorem 1. For all the questions, let a < b and let ƒ and g be bounded
functions on [a, b]. We won't repeat this every time. Do not assume
that any of the functions is integrable, unless specified: many of the in-
termediate results hold for non-integrable functions as well. In many of
the questions, you will need to use the results of one or various previous
questions in your proof.
(a) Prove that f is integrable on [a, b] if and only if Ve > 0, there exists
a partition P of [a, b] such that Up(f) – Lp(ƒ) < €.
Hint: you may need to use the fact about real numbers:
Let x = R. IF Ve > 0, x < € THEN x ≤ 0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff4893c46-a94c-4c81-a9fc-277ce07300a6%2Fec1f99f4-41e0-4ccb-b340-215ebdd75f1c%2F568bho5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The goal of this question is to prove the following result from the definition
of integral:
Theorem 1. Let a <b. Let f and g be bounded functions on [a, b].
If ƒ and g are integrable on [a, b]
Then fg is integrable on [a, b].
We will break the proof into pieces and guide you through them. In
mathematical terms, you will be proving a few "lemmas" before you prove
Theorem 1. For all the questions, let a < b and let ƒ and g be bounded
functions on [a, b]. We won't repeat this every time. Do not assume
that any of the functions is integrable, unless specified: many of the in-
termediate results hold for non-integrable functions as well. In many of
the questions, you will need to use the results of one or various previous
questions in your proof.
(a) Prove that f is integrable on [a, b] if and only if Ve > 0, there exists
a partition P of [a, b] such that Up(f) – Lp(ƒ) < €.
Hint: you may need to use the fact about real numbers:
Let x = R. IF Ve > 0, x < € THEN x ≤ 0.
![(b) Prove that if f is integrable
on
[a, b] then so is f².](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff4893c46-a94c-4c81-a9fc-277ce07300a6%2Fec1f99f4-41e0-4ccb-b340-215ebdd75f1c%2F4uu19ek_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(b) Prove that if f is integrable
on
[a, b] then so is f².
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