Infinite and infinitesimal

If I may, I would like to advert for a moment to the recent development of non-standard analysis and sketch how infinite and infinitesimal numbers can be presented. In this I follow a beautiful expository article of Ingleton (22) though, in my haste scarcely doing him, or Luxemburg on whom he leans, full justice. Consider all infinite sequences of real numbers X (x, x, ,xn, ) and let two such entities be equivalent if they differ only in a finite number of elements i.e., X E X if x =xf for all but a finite number of n. From now on we can consider the entity X to be the equivalence class and representable by any of its members just as the rational 1/2 is the class (1/2, 2/4, 3/6,..). We... 

The student should be able to discriminate between convergent and divergent series. I shall give tests very shortly. To simplify matters, it may be assumed that the series discussed in this work satisfy the tests of convergency. It is necessary to bear this in mind, otherwise we may be led to absurd conclusions. E. W. Hobson s On the Infinite and Infinitesimal in Mathematical Analysis, London, 1902, is an interesting pamphlet to read at this stage of our work. 

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