An Analysis of the First Proofs of the Heine-Borel Theorem - Lebesgue's Proof

Nicole R. Andre (Wittenberg University), Susannah M. Engdahl (Wittenberg University), and Adam E. Parker (Wittenberg University)

Lebesgue's Proof

In 1904, Lebesgue published his version of the theorem [14], which he said was due to Borel.

To compare the two numbers me, mi, we will use a theorem attributed to M. Borel:

If one has a family of intervals Δ such that any point on an interval (a,b), including a and b, is interior to at least one of Δ, there exists a family formed of a finite number of intervals Δ and that has the same property [any point of (a,b) is interior to one of them].

Note that where Lebesgue wrote (a,b) for a closed and bounded interval, we would write [a,b]. Unlike his predecessors, Lebesgue assumed the least upper bound property as his characterization of completeness.

In the passage shown below, Lebesgue started by presenting a new definition - that if [a,x] can be covered by a finite number of subintervals, then x is reached. In his notation, if x is reached, then so are all points between a and x. If x is not reached, then neither are any of the points between x and b (because if there were a y between x and b that was reached, then [a,y] would be covered by a finite number of subintervals, and so would [a,x]).

Let (á,β) be one of the intervals Δ containing a, the property to demonstrate is evident for the interval (a,x), if x is contained between á and β; I want to say that this interval may be covered with the help of a finite number of intervals Δ, which I express in saying that the point x is reached. It must be demonstrated that b is reached. If x is reached, all the points of (a,x) are [reached]; if x is not reached, none of the points of (x,b) are [reached].

He assumed that b is not reached (else the proof is done), then defined x0 to be the “first point not reached” or the “last point reached”. In modern notation, he defined x0 to be the greatest lower bound of the set \[X=\{x\in\left[a,b\right]\,\vert\,x\,\,{\rm is}\,\,{\rm not}\,\,{\rm reached}\}.\] This set is nonempty and bounded, and therefore has a greatest lower bound .

Now x0 is contained in some interval, which he called (á1,β1). In the following passage, he then chose two points x1 and x2 satisfying α1 < x1 < x0 < x2 < β1. By the definition of x0 he saw that x1 is reached and x2 is not reached. Because x1 is reached, [a, x1] is covered by a finite number of intervals. If we take that collection and append the interval (á1,β1) we get a finite collection that covers x2. This is a contradiction. Therefore b must have been reached.

Let x1 be a point of (á1,x0), x2 a point of (x01); x1 is reached by assumption, the intervals Δ in finite number which are used to reach it, plus the interval (á1, β1) allows x2 > x0; x0 is neither the last point reached, nor the last not reached; therefore b is reached (1).

In his footnote, Lebesgue explained Borel’s contributions. He mentioned that Borel required that the covering be countable, and noted that this may sometimes be adequate. However, he felt that the general theorem would be more useful.

(1) M. Borel gave, in his Thesis and in his Lessons on the theory of functions, two demonstrations of this theorem. These demonstrations essentially suppose that the set of intervals Δ are countable; this suffices in some applications; there is however interest in demonstrating the theorem of the text. For example, for the applications that I made in my Thesis of M. Borel’s theorem, it was necessary that he demonstrated for a set of intervals Δ having the power of the continuum.

Finally we give our overview of Lebesgue’s proof.

  • Completeness in the form of existence of the supremum for every non-empty bounded set will be required to carry out this proof.
  • This proof is very short and is particularly easy to follow. In fact, starting the proof may lead to an “ah-ha!” moment where the students can complete the necessary steps.
  • It appears that those textbooks that don’t use the “divide and conquer” technique of Cousin do use Lebesgue’s method. This proof may integrate particularly nicely into those courses.
  • This technique of proof is very useful, and appears, for example, in the intermediate value theorem. If students have not already seen it, it is likely that they will.
  • This proof works just as well for countable as uncountable covers.
  • The proof is non-constructive. There is no way that we can ascertain the finite covering by working the proof.

This is the one! The proof is thoroughly modern and simple to follow. In comparison, all previous arguments are cumbersome and overly complicated. It is no wonder that many people choose to attach Lebesgue’s name to Borel’s when referencing the theorem. Certainly this proof should be presented in any real analysis course, and probably in many others!

Henri Lebesgue (1875-1941) (Convergence Portrait Gallery)