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 x
2 < 2} and B = {x ? Q : x > 0 and x
2 > 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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