I've been trying to solve a kinda long problem for a few days and can't quite finish it.
Prove the equivalence between
(1) Every bounded sequence has a convergent subsequence.
(2) All cauchy sequences are convergent.
(3) If $ (I_n)_{n\geq 1} $ is a family of closed nested intervals so that $|I_n| \rightarrow 0 $ as $n \rightarrow \infty$ there exist a unique $x$ so that $x \in \bigcap_{n=1}^\infty I_n$.
(4) Every upper bounded non-empty set has a supremum.
(5) Every monotone, upper bounded sequence has a supremum.
I managed to get $(1)\rightarrow(2)\rightarrow(3)$ and $(4)\rightarrow(5)\rightarrow(1)$ so I obviously would love to prove $(3)\rightarrow(4)$.
I manage to get a decreasing sequence since if there's no supremum then for every upper bound $s$ there exist at least one $\varepsilon > 0$ so that $s – \varepsilon$ is also an upper bound. So I can choose one upper bound $s$ and consider the sequence $(s – \varepsilon_n)_{n \in \mathbb{N}}$ that gets closer to the set as $n$ grows, where $s – \varepsilon_n$ is an upper bound for all $n$.
Now I'd like to construct a monotonically increasing sequence of elements of the set so that $|A_n – (s – \varepsilon_n)| \rightarrow 0$ but I'm not sure how to do this, or if my decreasing sequence is convenient for this, it does seem quite ugly to work with, but couldn't really come up with an improvement.
I've consider taking another direction and proving that (3) implies something else and that in turn that implies (4), but I couldn't find a good way of doing it. Maybe I missed something?
Any help would be greatly appreciated.
Best Answer
You're on the right track. Assuming there's no sup, you deduce that the set likewise has no maximum element. Therefore, for each $n$, you can choose an element $x_n$ of the set and an upper bound $y_n$ with $y_n-x_n<1/n$. You can also stipulate that $\{x_n\}$ is increasing and $\{y_n\}$ is decreasing.