I finally proved this on my own for the first time, though I admit that I read it once over a month ago, superficially following it while afterwards sort of struggling to re-construct it. Partly through this, I realized that my point-set topology foundation was insufficient, after which I wrote  and .
Definition 1 For a topological space on , a subset is compact iff every open cover of it has a finite subcover.
Bolzano-Weierstrass Theorem Every sequence in a closed, bounded subset of has a convergent subsequence.
Proof: I’ve proved this myself for , by bisecting intervals and using the nested interval theorem in some unreleased notes. The result is easily generalizable to .
On , which is a metric space for which the metric is given by
we shall use the topology generated by sets of the form , where is the Euclidean norm associated with the metric. We call these rational open balls, centered at with radius and denote via .
Heine-Borel Theorem Closed and bounded in is equivalent to compact.
Proof: Assume is compact, which means that every open cover of it has a finite subcover. Apply this to any arbitrary covering of by rational open balls, and since a finite subcover exists, the subcover is bounded. Suppose by contradiction that is not closed. Then there exists a sequence in converging to a point . At every point in the sequence, we can take an open ball containing it but not . At every point not covered after this, we cover it with an open ball that does not contain to obtain an open cover of . Because no element of the cover contains , by the fact that Hausdorff, it is apparent there is no element in the cover that contains infinitely many points in the sequence converging to . Thus any finite subset of the cover, contains only a finite number of elements in that sequence, which means there is no finite subcover. This means that a compact subset of must be closed.
Now assume is closed and bounded in . With for the rational point and radius a countable sequence, we have a sequence of all rational open balls, This sequence of sets obviously covers . Suppose by contradiction that there is no finite subcover here. Then for all , there exists such that By the Bolzano-Weierstrass theorem, has a convergent subsequence that converges to a value , since is closed. No can contain , because if did for some , then each element of the convergent subsequence past some point would be contained by the union of balls corresponding to it as given in , a contradiction. Because covers though, every has a smallest natural number such that . But this contradicts that is not contained by any of the s. Thus, closed and bounded implies that the covering by all rational open balls has a finite subcover. For any open cover, every element in the cover is equal to some union of rational open balls. Every open cover corresponds in this way to a covering by a collection of rational open balls , each of which is contained in some element of the original open cover . That covering by rational open balls we’ve already established has a subcover. Since every element in this subcover is contained by some element of , is also guaranteed to have an open cover.
As a side note, I am feeling somewhat enticed into mathematical logic and computability theory. At the same time, I fear that maybe I am wasting too much time on something as impractical as pure math that in addition to impracticality has already mostly reached its bottleneck. Maybe instead I should be learning or doing some more practical things. My original plan was to prove Alexander subbase theorem and Tychonoff’s theorem on my own, and of course, before I do that, I should prove Heine-Borel theorem on my own, which I just did. However, a short diversion into Cantorian set theory (see ) brought me to read casually about mathematical logic related matters. I finally learned the difference between propositional logic (no quantifiers), predicate calculus or first-order logic (quantifiers over sets), and higher-order logic (quantifers over sets and functions), which I don’t think I will ever forget. As for the word “predicate”, its etymology is prae = before, dicare = to dictate; this makes sense since the predicate of a sentence in essence gives a verdict of true or false for something. One can interpret “predicated upon”, which means “based upon”, as making a verdict based upon something.
I was reminded of the halting problem, the proof of which I’ve read before but could never remember. I also saw the names of Russell, Skolem, Lowenheim, Tarski, Godel, Turing, Church, and I thought how great it would be if I could actually understand some of their work. I was also reminded of the Chinese (and later naturalized American) logician Hao Wang, who was actually the advisor of Stephen Cook, the man celebrated for the Cook-Levin theorem. He studied at 西南联大 during World War II, like Steve Hsu’s father. Interestingly, Tarski had a Chinese student named Chen Chung Chang (张晨钟), who was born in 1927, who as a professor at UCLA wrote the standard textbook on model theory with Jerome H. Keisler, another student of Tarski. That the Polish Tarski later became a professor at Berkeley also reminded me of Thomas Scanlon, a 1970s born model theorist at the same institution. Another logician I stumbled upon was Paul Bernays.