Classical ideas about principles of the theory of limits

The article presents limit theorems that have a conditional character. We provide not only full justification of the theory of limits principles based on the previously developed real numbers theory, but also the evidence that these principles are equivalent. It is shown that all the four principles are equally applicable for proving the general analysis theorems. In order to solve the problems on limits of sequences the Cauchy, Weierstrass and Cantor principles are adjusted better than the Dedekind principle. The Cantor and Dedekind theorems are proved for the number scale, where by the points are meant the numbers, and by the segments – some sum-total numbers. The statements of the theorems are understood literally, as referring to the ordinary segments and points on the geometrical rather than on the number scale. These statements are the consequences of the definitions and axioms accepted in geometry.

References

[1] Sidnyaev N.I., Krylov D.A. Nepreryvnost’. Beskonechno malye i beskonechno bol’shie funktsii [Continuity. Infinitely-small and infinitely-large functions]. Moscow, Bauman Press, 2014, 38 p.

[2] Sidnyaev N.I., Nevskiy Yu.A., Sadykhov G.S. Beskonechno malye i beskonechno bol’shie: teoriya i praktika [Infinitely-small and infinitely-large: theory and practice]. Moscow, Bauman Press, 2015, 22 p.

[3] Natanson I.P. Teoriya funktsiy veshchestvennoy peremennoy [Theory of functions of real variable]. Sankt-Peterburg, Lan’, 2008, 560 p.

[4] Sidnyaev N.I., Gordeeva N.M., Popushina E.S., Rybdalova O.D. Rukovodstvo k resheniyu zadach po vektornomu analizu [Guidance to solving problems of vector analysis]. Moscow, Bauman Press, 2015, 62 p.

[5] Sidnyaev N.I., Sobolev S.K. Mechanisms for improvement of mathematical education in technical university. Alma Mater (Vestnik vysshey shkoly) [Alma Mater (High School Herald)], 2015, no. 6, pp. 5–14.

[6] Sidnyaev N.I., Tomashpol’skiy V.Ya. O matematike, matematikakh i kafedre “Vysshaya matematika” [On mathematics, mathematicians and “Advanced mathematics” chair]. Moscow, Bauman Press, 2014, 258 p.

[7] Fikhtengol’ts G.M. Kurs differentsial’nogo i integral’nogo ischisleniya. T. 1 [Differential and integral calculation course. Vol. 1]. Moscow, Fizmalit publ., 2003, 680 p.

[8] Pis’mennyy D.T. Konspekt lektsiy po vysshey matematike. Ch. 1 [Lecture notes on advanced mathematics]. Moscow, Ayris-press publ., 2007, 282 p.

[9] Smirnov V.I. Kurs vysshey matematiki. T. 1 [Advanced mathematics course. Vol. 1]. Sankt-Peterburg, BKhV-Peterburg publ., 2008, 624 p.

[10] Kudryavtsev L.D., Kutasov A.D., Chekhlov V.I., Shabunin M.I. Sbornik zadach po matematicheskomu analizu. T. 1. Predel. Nepreryvnost’. Differentsiruemost’ [Problem book on mathematical analysis. Vol. 1. Limit. Continuity. Differentiability]. Moscow, Fizmatlit publ., 2003, 362 p.

[11] Il’in V.A., Poznyak E.G. Osnovy matematicheskogo analiza. Ch. 1 [Mathematical analysis fundamentals. P. 1]. Moscow, Fizmatlit publ., 2005, 648 p.