Proof of nested interval theorem from bolzano weiertress theorem

Question

I am able to proof bolzano weiertress theorem from nested interval theorem but can I do the reverse part?

Answer 1

Yes you can suppose you have $I_1=[\alpha_1,\beta_1]\supset I_2=[\alpha_2,\beta_2] \supset ... \supset I_n=[\alpha_n,\beta_n] \supset ...$

Then take the sequence $\alpha_1 \leq \alpha_n \leq b_1$ which is bounded , from B-W there is a subsequene $\alpha_{k_{n}} \to \alpha $ and show now that this $\alpha$ belongs to $\bigcap_{n=1}^{\infty} I_n$.

Set $n \in \Bbb N$ , then for $m\geq n$ you have $k_{m}\geq k_{n} \geq n$. So $\alpha_n \leq \alpha_{k_{m}} \leq \beta_{k_{m}} \leq \beta_n$. So if you take $m\to \infty $ you get $\alpha_n\leq \alpha \leq \beta_n$ which means that $\alpha \in [\alpha_n,\beta_n]$.