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Fundamental Properties of Supremum and Infimum, Including sup(A + B) = sup(A) + sup(B)
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Real Analysis Study Help for Baby Rudin, Part 1.4. After reviewing the definition of supremum and infimum, I do three things: 1) work through the proof of Theorem 1.11 in Walter Rudin's textbook (Baby Rudin), "Principles of Mathematical Analysis" (about showing ordered sets with the least upper bound property also have the greatest lower bound property), 2) solve exercise 5 from Chapter 1 in Baby Rudin (show that A is a nonempty set of real numbers which is bounded below, then -A is bounded above and sup(-A) = -inf(A), and 3) prove that the supremum is an additive set function on subsets of the real numbers that are bounded above: sup(A + B) = sup(A) + sup(B).
#realanalysis #supremum #supandinf
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⏱️TIMESTAMPS⏱️
(0:00) Introduction and Infinity is Really Big article.
(1:12) Definition of supremum and examples.
(3:19) Least upper bound property and greatest lower bound property.
(4:13) Every set with the LUB property has the GLB property (Theorem 1.11 in Baby Rudin).
(7:35) Bonus: Solve exercise 5 in Chapter 1 of Baby Rudin.
(11:36) sup(A + B) = sup(A) + sup(B) (the supremum is an additive set function on the set of all subsets of the real numbers which are bounded above).
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#realanalysis #supremum #supandinf
Other Links and resources
===============================
⏱️TIMESTAMPS⏱️
(0:00) Introduction and Infinity is Really Big article.
(1:12) Definition of supremum and examples.
(3:19) Least upper bound property and greatest lower bound property.
(4:13) Every set with the LUB property has the GLB property (Theorem 1.11 in Baby Rudin).
(7:35) Bonus: Solve exercise 5 in Chapter 1 of Baby Rudin.
(11:36) sup(A + B) = sup(A) + sup(B) (the supremum is an additive set function on the set of all subsets of the real numbers which are bounded above).
AMAZON ASSOCIATE
As an Amazon Associate I earn from qualifying purchases.
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