This question is from abbott's Understanding Analysis:
Start with the Bolzano--Weierstrass Theorem and use it to construct a proof of the Nested Interval Property.
My try: Let $I_n=[a_n,b_n]$ be closed intervals such that $I_1 \supseteq I_2 \supseteq \dots$. So here $a_n$ is a (nondecreasing) bounded sequence, so it contains a convergent subsequence by hypothesis, so does $b_n$.
But here I am stuck. I have temptation to use Monotone Convergence Theorem, but that would imply that I am assuming Axiom of Completeness, or worse, Nested Interval Property itself!