Unformatted text preview: x < ± for every ± > 0, then x ≤ 0. (8) Complete the partial discussion in Rudin’s Step 6 (pp. 1920) establishing multiplication and its properties in R + . (9) Let E and F be nonempty subsets of R . Suppose that for all x ∈ E and y ∈ F we have x < y . Prove that sup E ≤ inf F. Can “ ≤ ” be replaced by “ < ”? (10) Is the set α = { x ∈ Q : x 2 < 3 } a Dedekind cut? If so, prove that α 2 = 3. If not, ﬁnd a Dedekind cut whose square is 3. (11) Theorem 1.20 in Rudin is proven “abstractly” as a consequence of the leastupperbound property. Give a “concrete” proof using the construction of R by Dedekind cuts. 1...
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 Spring '08
 EduardoFaingold
 Microeconomics, Dedekind cut, nice iteration scheme

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