Abstract
IN this book, the theory of convergence is developed on two fundamental assumptions. The first of these is concerned with upper bounds, namely, that a certain set of numbers has in it a least number ; while the second refers to irrational number as the limit of a sequence of rational numbers, namely, that every irrational number is the limit of a monotonic increasing sequence of rational numbers. With the aid of these assumptions, the theory of convergence is developed without recourse to the properties of Dedekind cuts. The 'real' number appears only in the appendix, where the assumptions used in the body of the work are proved to be consequences of the definition of real number. In the appendix also, the first of the above-mentioned assumptions appears as a theorem, and a proof of the second is given. In fact, the appendix contains as much of the foundations of analysis as is necessary to justify the assumptions made in the initial chapters of the book. A short historical survey prefacing an examination of these 'foundations' shows why such a complex structure as the Dedekind cut is essential to the definition of number.
A Text-Book of Convergence
By W. L. Ferrar. Pp. vii + 192. (Oxford: Clarendon Press; London: Oxford University Press, 1938.) 10s. 6d. net.
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A Text-Book of Convergence. Nature 142, 556 (1938). https://doi.org/10.1038/142556a0
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DOI: https://doi.org/10.1038/142556a0