Exercise 2.6.6. (a) Assume the Nested Interval Property (Theorem 1.4.1) is
true and use a technique similar to the one employed in the proof of the Bolzano– Weierstrass Theorem to give a proof for the Axiom of Completeness. (The reverse implication was given in Chapter 1. This shows that AoC is equivalent to NIP.)
(b) Use the Monotone Convergence Theorem to give a proof of the Nested
Interval Property. (This establishes the equivalence of AoC, NIP, and MCT.)
(c) This time, start with the Bolzano–Weierstrass Theorem and use it to
construct a proof of the Nested Interval Property. (Thus, BW is equivalent to NIP, and hence to AoC and MCT as well.)
(d) Finally, use the Cauchy Criterion to prove the Bolzano–Weierstrass The-
orem. This is the ﬁnal link in the equivalence of the ﬁve characterizations of completeness discussed at the end of Section 2.6.
This is from Stephen Abbott's "Understanding Analysis". I've attached chapter 2 from the book.