Question

Recall the following theorem, phrased in terms of least upper bounds.
Theorem (The Least Upper Bound Property of R). Every nonempty subset of R that
has an upper bound has a least upper bound.
A consequence of the Least Upper Bound Property of R is the Archimedean Property.
Theorem (Archimedean Property of R). For any x; y 2 R, if x > 0, then there exists
n 2 N so that nx > y.
Prove the following statements by using the above theorems.
(a) For any two real numbers a; b 2 R, if a < b, then there exists a real number r 2 R
such that a < r < b.
(b) Prove that for any two rational numbers a; b 2 Q, if a < b, then there exists an
irrational number r 2 R, r =2 Q, such that a < r < b.
(c) For any two real irrational numbers a; b 2 R, a; b =2 Q, if a < b, then there exists
a rational number q 2 Q such that a < q < b.
(d) Prove that the Least Upper Bound Property is equivalent to the Greatest Lower
Bound Property: \Every nonempty subset of R that has a lower bound has a
greatest lower bound."

Answer 1