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section4and5notes - Lectures 4 and 5 • Define maximum...

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Unformatted text preview: Lectures 4 and 5 • Define maximum and minimum of a set. Define a set being bounded above, bounded below, and bounded. Finite sets always have maximums and minimums, and are always bounded. Infinite sets may not have maximums or minimums, and may not be bounded. Ex: {r ∈ Q : r2 < 2}, and {a ∈ Q : r > 2}. • Define supremum and state the completeness axiom. • Define R using our field axioms and the completeness axiom. • Prove that the completeness axiom implies infimums exist for sets that are bounded below. • Prove the Archimedean property of the real numbers. • Prove the density of Q in R. The irrational numbers are also dense in R (homework problem). • Define the set R ∪ {−∞, ∞}. Stress that this set has order structure, but no algebraic structure. • Extend the notion of supremum to the set R ∪ {−∞, ∞}. Then supremums and infimums exist for any non-empty subset of R. 1 ...
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