Sequences and Limits

Wolfgang Mueckenheim^*

University of Applied Sciences, Augsburg, Germany.

DOI: 10.4236/apm.2015.52007                

Abstract

It is widely held that irrational numbers can be represented by infinite digit-sequences. We will show that this is not possible. A digit sequence is only an abbreviated notation for an infinite sequence of rational partial sums. As limits of sequences, irrational numbers are incommensurable with any grid of decimal fractions.

Keywords

Series, Sequences, Limits, Definability of Real Numbers, Set Theory

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Mueckenheim, W. (2011) Mathematik für die ersten Semester. 3rd Edition, Oldenbourg Verlag GmbH, Muenchen, 193. http://www.amazon.de/Mathematik-f%C3%BCr-die-ersten-Semester/dp/348670821X/ref=sr_1_2?s=books&ie=UTF8&qid=1400566108&sr=1-2&keywords=Mathematik+f%C3%BCr+die+ersten+Semester
[2]	Cantor, G. (1889) Bemerkungen mit Bezug auf den Aufsatz: Zur Weierstra?-Cantorschen Theorie der Irrationalzahlen. Mathematische Annalen, 33, 476. http://dx.doi.org/10.1007/BF01443973
[3]	Cantor, G. (1891) über eine elementare Frage der Mannigfaltigkeitslehre. Jahresbericht der Deutschen MathematikerVereinigung, 1, 75-78.
[4]	Mueckenheim, W. (2008) The Infinite in Sciences and Arts. In: Sriraman, B., Michelsen, C., Beckmann, A. and Freiman, V., Eds., Proceedings of the 2nd International Symposium of Mathematics and Its Connections to the Arts and Sciences (MACAS2), Centre for Science and Mathematics Education, University of Southern Denmark, Odense, 265-272. http://static.sdu.dk/mediafiles//Files/Om_SDU/Centre/C_NAMADI/Skriftserie/MACAS_samlet.pdf%