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!
Best Answer
As you noted, there is a subsequence $n_k$ such that $a_{n_k}\to a$. As $I_n$ is closed and contains almost all $a_{n_k}$, we have $a\in I_n$. As this holds for all $n$, $a\in\bigcap I_n$