A. Please hand in the following problems from Taylor's Foundations of Analysis.
21[ 8, 11 ],
27[ 4, 8 ].
B. Please hand in the following additional problem.
Prove that if x < y are two real numbers then there is a rational number p and an irrational number q such that x < p < q < y.
The Well Ordering Principle for the natural numbers says that every nonempty subset S of N has a least element. It is a consequence of the Peano axioms (see 15[ 17 ].) Show that for every real number x > 1 which is not an integer, there is a natural number n ∈ N such that n < x < n + 1.