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 nonempty subset of R. 1 ...
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 Winter '11
 Wiley
 Differential Equations, Equations, Sets, Empty set, completeness axiom

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