In his 1895 “Sur les fonctions de n variables complexes,” Pierre Cousin [7] extended the Heine-Borel Theorem to arbitrary covers. On page 22, he wrote:
Define a connected space S bounded by a simple or complex closed contour; if to each point of S there corresponds a circle of finite radius, then the region can be divided into a finite number of subregions such that each subregion is interior to a circle of the given set having its center in the subregion.
Before we examine the proof, we mention a few points:
Every nested sequence of closed, bounded sets in \({\mathbb R}^2\) has a non-empty intersection.
Cousin started immediately with a description of his proof technique, just as we teach our students to do, and proceeded by contradiction. He used the “divide and conquer” technique that probably appears elsewhere in a real analysis course.
Diagram 2: In this diagram we see the “divide and conquer” method that Cousin employed in his proof.
Cousin took the region S and divided it into \(n>1\) subregions (\(n\) an integer). If the entire region required an infinite number of circles to cover it, then at least one of the subregions must also require an infinite number of circles to cover it. Cousin called that subregion S_{1}. It is the light grey, lower right quadrant of Diagram 2.
We suppose, in fact, the lemma is false: we divide S into squares using parallels to the coordinate axes, in a way that the number of obtained regions is at least equal to a certain integer n; there is at least one of these regions S_{1}, for which the lemma is still false.
Cousin then iterated the process, dividing S_{1} into n sub-subregions. At least one of these sub-subregions must require an infinite number of circles to cover it. He called this region S_{2} (the slightly darker grey, lower left quadrant of S_{1} in Diagram 2.) By continuing in this fashion, he created an infinite sequence of nested closed square regions S_{1 }, S_{2 }, … , S_{p }, … (the successively darker grey regions in Diagram 2.) He then applied the nested interval property to show there is a point M that is common to each of these S_{p}, and that M is in the interior or boundary of S.
Subdividing S_{1 }into squares and portions of squares in number at least equal to n, I deduce S_{2}, in the same way that S_{1 }is deduced from S; in following the reasoning, I arrive at an indefinite series of squares or portions of squares S_{1 }, S_{2 }, … , S_{p }, … ; it is clear that S_{p}, for p increasing indefinitely, has for a limit a point M interior to S or on its perimeter; ...
Because M is in each S_{p}, and each S_{p} cannot be covered by a finite number of circles, certainly each S_{p} cannot be covered by a single circle. However, as Cousin noted, this is impossible, because M is in S and so some circle of positive radius covers it (the blue circle in Diagram 2). That circle will contain some S_{n} and all subsequent S_{n+k}, which is a contradiction to the way that the S_{p} were constructed.
... one arrives at this conclusion that one can find a square S_{p }surrounding M or adjacent to M which is not contained in the interior of one of the circles of the statement; however this is impossible because to the point M corresponds a circle of finite radius having this point for a center.
The following may be helpful when considering using Cousin’s proof in a class.
We believe that this proof would be particularly applicable if the Heine-Borel Theorem is being taught in a point-set topology course or in an advanced calculus class where domains of functions are often two-dimensional or higher.