Real analysis

HW2: MA 34100. DUE 1/25.

A Dedekind cut is a pair (A, B) where A, B ⊂ Q such that: (a) A 6= ∅, B 6= ∅; (b) If a ∈ A and c < a, then c ∈ A; (c) If b ∈ B and c > b, then c ∈ B; (d) If b /∈ B and a < b, then a ∈ A; (e) If a /∈ A and b > a, then b ∈ B; (f) For each a ∈ A there is some b > a so that b ∈ A; (g) For each b ∈ B there is some a < b so that a ∈ B;

1. Let A = {x ∈ Q : x < 0 or x2 < 2} and B = {x ∈ Q : x > 0 and x2 > 2}. Prove that (A, B) is a Dedekind cut.

2. Let (A1, B1) and (A2, B2) be Dedekind cuts and define A3 := {a1 + a2 : ai ∈ Ai, i ∈ {1, 2}}, B3 := {b1 + b2 : bi ∈ Bi, i ∈{1, 2}}. Prove that (A3, B3) is a Dedekind cut.

3. Let X, Y be non-empty sets and f : X ×Y → R have bounded range. Define f1(x) := sup{f(x, y) : y ∈ Y}, f2(y) := sup{f(x, y) : x ∈ X}.

Show that

sup{f(x, y) : x ∈ X, y ∈ Y} = sup{f1(x) : x ∈ X} = sup{f2(y) : y ∈ Y}.

4. Let f and f1 be as in Problem 3 and define g2(y) := inf{f(x, y) : x ∈ X}. Prove that sup{g2(y) : y ∈ Y}≤ inf{f1(x) : x ∈ X} and show via an example that strict inequality is possible. (The shorthand for this inequality is: supy infx f(x, y) ≤ infx supy f(x, y).)

5. Let X be a non-empty set and suppose f : X → R has bounded range. If a ∈ R, show that

sup{a + f(x) : x ∈ X} = a + sup{f(x) : x ∈ X} and

inf{a + f(x) : x ∈ X} = a + inf{f(x) : x ∈ X}.

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