Prop 7.7) Let f [a, b] R then, 1) L(f, P) ≤U(f, P) 2) Let P₁ and P2 be partitions of [a, b], then the following holds, L(f, P₁) ≤ U(f, P₂)
Prop 7.7) Let f [a, b] R then, 1) L(f, P) ≤U(f, P) 2) Let P₁ and P2 be partitions of [a, b], then the following holds, L(f, P₁) ≤ U(f, P₂)
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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![Prop 7.7)
Let f [a, b] R then,
1) L(f, P) ≤U(f, P)
2) Let P₁ and P2 be partitions of [a, b], then the following holds, L(f, P₁) ≤ U(f, P₂)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fecc418e3-a973-4d51-90e0-10d2ad078b59%2Fc33e60f2-6644-4def-a1d2-6122d84c4351%2Fwyc9sc_processed.png&w=3840&q=75)
Transcribed Image Text:Prop 7.7)
Let f [a, b] R then,
1) L(f, P) ≤U(f, P)
2) Let P₁ and P2 be partitions of [a, b], then the following holds, L(f, P₁) ≤ U(f, P₂)
![Exercise 3 Suppose A and B are non-empty subsets of the real numbers such that for
any x EA and y B, we have x ≤y. Prove that sup(A) ≤ inf(B).
Exercise 5
Complete all the details of Proposition 7.9.
Let f [a, b] R be a bounded function; say m≤ f(x) ≤ M for all x [a, b]. Then we
have
m(b − a) ≤ L(f) ≤ U(ƒ) ≤ M(b − a)
You will need to use Proposition 7.7 and Exercise 3 above to show that L(f) ≤U(f)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fecc418e3-a973-4d51-90e0-10d2ad078b59%2Fc33e60f2-6644-4def-a1d2-6122d84c4351%2Fwjn87j8_processed.png&w=3840&q=75)
Transcribed Image Text:Exercise 3 Suppose A and B are non-empty subsets of the real numbers such that for
any x EA and y B, we have x ≤y. Prove that sup(A) ≤ inf(B).
Exercise 5
Complete all the details of Proposition 7.9.
Let f [a, b] R be a bounded function; say m≤ f(x) ≤ M for all x [a, b]. Then we
have
m(b − a) ≤ L(f) ≤ U(ƒ) ≤ M(b − a)
You will need to use Proposition 7.7 and Exercise 3 above to show that L(f) ≤U(f)
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