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An Analysis of the First Proofs of the Heine-Borel Theorem - Conclusion

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


The use of primary sources or historical documents can be a powerful teaching tool in any course. It can provide context and improve retention of the material.

The Heine-Borel Theorem will be familiar to any math major, though many may struggle with the abstractness of the proof. By presenting the historical development along with the mathematics, we may help students understand the theorem. Students often are surprised to learn that this theorem escaped many brilliant mathematicians for decades. They also may enjoy the debate surrounding the name.

Of course, our goal is not merely to provide historical anecdotes. By translating and analyzing historical proofs, we provide teachers with tools to enrich their presentation of this topic in a variety of ways. A teacher may either augment or replace a proof from the textbook. Or projects for students can be developed. Regardless, we believe this information can improve the teaching of this topic.

We hope that readers find this a valuable resource in their classes.



The authors would like to thank Dr. Tim Bennett for his help with their translations. They would also like to thank the referees and especially the editor of Convergence for their many helpful suggestions, as well as their patience, in improving this paper.


About the Authors

Nicole Andre is currently a medical student at Ohio University Heritage College of Osteopathic Medicine. She earned her B.S. in Biology at Wittenberg University in Springfield, Ohio, in May of 2012. Along with studying biology she focused on achieving proficiency in the German language and studied for a semester in Marburg, Germany. Upon completing her medical education, Nicole hopes to work as a physician in an underserved area.

Susannah Engdahl is a senior at Wittenberg University, where she is pursuing a major in physics and minors in mathematics and computational science. After leaving Wittenberg, she plans to attend graduate school to study biomedical engineering.

Adam Parker is associate professor of mathematics at Wittenberg University. He has undergraduate degrees in mathematics and psychology from the University of Michigan, and earned his mathematics Ph.D. in 2005 from the University of Texas at Austin under the direction of Dr. Sean Keel. His research interests include algebraic geometry and the history of mathematics.

Nicole R. Andre (Wittenberg University), Susannah M. Engdahl (Wittenberg University), and Adam E. Parker (Wittenberg University), "An Analysis of the First Proofs of the Heine-Borel Theorem - Conclusion," Loci (August 2013), DOI:10.4169/loci003890