I tried to prove two directions.
The ($\Leftarrow$) direction is not so hard, it's just in $\mathbb R$, since $\mathbb R$ is complete, Heine-Borel Theorem holds. However, I have a hard time proving ($\Rightarrow$) direction. Which is using Heine-Borel Theorem we can prove the Completeness Axiom.
There is a similar problem posted on this website with solutions. However, it's hard for me to understand either since I have no background on topology…Least Upper Bound Axiom for Heine-Borel Theorem
Any help? Thanks~
Best Answer
We show if Heine-Borel theorem holds on $\mathbb{R}$, then $\mathbb{R}$ has the least upper bound property.
Let $F$ be a bounded subset of $\mathbb{R}$, we want to show that $F$ has a least upper bound.
Note that the clousure $\overline F$ is closed and bounded.
If $\overline F$ has a maximal element, we can show this element is actually the least upper bound for $F$, using the fact that the maximal element is in $\overline F$.
Else, by Heine-Borel theorem the open cover of $\overline F\subset\cup_{x\in \overline F}(-\infty, x)$ has a finite sub cover, that is $$F\subset (-\infty, x_1) \cup (-\infty, x_2)\cup \cdots \cup (-\infty, x_n)$$ where each $x_i$ are in side the clousure $\overline F$
Call the largest $x_i$, "$k$", then $k$ is the least upper bound of $F$. You can try to show this using the fact that $k \in \overline F$, so every open ball of radius $\epsilon$ around $k$ contains some element of $F$.