Question

(1) In Section 1.4 we used the Axiom of Completeness (AoC) to prove the Archimedean Property of R (Theorem 1.4.2). Show that the Monotone Convergence Theorem can also be used to prove the Archimedean Property without making any use of AoC.

(2) Use the Monotone Convergence Theorem to supply a proof for the Nested Interval Property (Theore 1.4.1) that doesn't make use of AoC. These two results suggest that we could have used the Monotone Con- vergence Theorem in place of AoC as our starting axiom for building a proper theory of the real numbers

Answer 1

Both Answers are in the following attachment (a) and (b)

Explanation: