Introduction
In his 1895 “Sur quelques points de la théorie des fonctions,” Émile Borel (18711956) stated for the first time the essential result that has come to be known as the HeineBorel Theorem in the field of analysis [5, p. 52].
Here is the theorem: If one has on a straight line an infinite number of partial intervals, such that any point on the line is interior to at least one of the intervals, one can effectively determine a LIMITED NUMBER of intervals chosen among the given intervals and having the same property (any point on the line is interior to at least one of them).
Passage 1: The first statement of the HeineBorel Theorem, along with a translation.
Today we would state this half of the HeineBorel Theorem as follows.
HeineBorel Theorem (modern): If a set \(S\) of real numbers is closed and bounded, then the set \(S\) is compact. That is, if a set \(S\) of real numbers is closed and bounded, then every open cover of the set \(S\) has a finite subcover.
Students sometimes struggle with the HeineBorel Theorem; the authors certainly did the first time it was presented to them. This theorem can be hard to motivate as the result is subtle and the applications are not obvious. Its uses may appear in different sections of the course textbook and even in different classes. Students first seeing the theorem must accept that its value will become apparent in time. Indeed, the importance of the HeineBorel Theorem cannot be overstated. It appears in every basic analysis course, and in many pointset topology, probability, and set theory courses. Borel himself wanted to call the theorem the “first fundamental theorem of measuretheory” [6, p. 69], a title most would agree is appropriate.
In addition to its mathematical significance, the HeineBorel Theorem has a complex history. Theophil Hildebrandt (18881980) in [11, p. 424] stated, “As in the case of other important mathematical results, the conception of the Borel Theorem is an interesting chapter in mathematical history.” Broadly speaking, the story of the HeineBorel Theorem has two chapters: pre1895 and post1895. In 1989, Pierre Dugac examined letters between the key players and unraveled much of the narrative in his aptly titled “Sur la correspondance de Borel et le théorème de DirichletHeineWeierstrassBorelSchoenfliesLebesgue” [8]. Of particular interest will be how the theorem got its name given that Eduard Heine (1821–1881) died when Borel was only ten years old.
As with many results, people implicitly used the HeineBorel Theorem for decades before Borel published it in 1895. David Bressoud noted, “There are two immediate corollaries of the HeineBorel Theorem that are historically intertwined. They predate Borel’s Theorem of 1895” [6, p. 66]. Bressoud was referring to the BolzanoWeierstrass Theorem and the theorem that every continuous function on a closed and bounded interval is uniformly continuous (sometimes called Heine’s Theorem), but there were many others as well. Understanding pre1895 events will help us enrich the classroom experience and empathize with our students’ difficulties. After all, if the importance of the theorem escaped some of the great 19^{th} century analysts, we can forgive our students’ struggles.
In the post1895 chapter, we see a proliferation of proofs of the HeineBorel Theorem. Pierre Cousin, William Henry Young, Arthur Schoenflies, and Henri Lebesgue all published proofs within the next nine years. Obviously the later authors and journal editors either were unaware of the earlier publications or were convinced that their arguments improved significantly on what was already in the literature. Up to now, no comparative analysis of the mathematical contributions of these different proofs has been performed. This is the primary goal of our paper.
The story is further complicated because some of the aforementioned five mathematicians published multiple proofs. For example, we know that Borel’s first proof appeared in 1895. From Dugac [8] we know Borel wrote a very similar version in 1898 [3] and two more in 1903 [2] [4]. In our paper, we will concentrate on the first proofs by these authors but will acknowledge subsequent writing when appropriate.
The mathematics of these first proofs is interesting for many reasons. Some have limitations to their usage, whereas others are valid in more general situations. Some are in different dimensions. Some are constructive, and some are purely formal. Some are well written, and others leave out details. We will make particular note of how the different proofs relate to completeness of the real numbers as this connection is indispensable. Recall that the set of real numbers is complete because it has the following equivalent properties [6, p. 62], [1, p. 420]:

Every sequence of closed, nested intervals has a nonempty intersection.

Every bounded subset has a least upper bound.

Every Cauchy sequence converges.

Every infinite bounded subset has a limit point (BolzanoWeierstrass property).

Every monotonic bounded sequence converges (monotone convergence property).
In a modern real analysis course, the second property usually is taken as the definition of completeness, and completeness taken as part of the definition of the real number system. The remaining four properties are then proved as theorems. However, only one of the five mathematicians whose proofs we will discuss used the second property as his definition of completeness. The characterization of completeness that each author assumed is a key difference among the five proofs.
Just as knowing the history can enrich our classroom by shedding light on the difficulties of our students, understanding the mathematics in these proofs can be a valuable teaching tool. So in addition to unraveling the history, our goal is to provide a resource for teachers that will enable them to motivate the study of this essential theorem. This may be achieved in several ways. The teacher may recognize that the proof from her modern textbook is actually from one of these first papers. She can then provide the historical background for what is presented in her text. The instructor may also find that one of these proofs can be more fully integrated into his course than the one he is currently presenting. We hope that he feels free to use these proofs to replace or augment his current presentation. Finally, it makes an excellent project to have students examine a proof that is different from the one presented in their class. Indeed, this paper grew out of just such a project.