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Arzelà’s bounded convergence theorem

  • Amar Sarić EMAIL logo
Veröffentlicht/Copyright: 9. August 2023
Analysis
Aus der Zeitschrift Analysis Band 44 Heft 2

Abstract

This article presents a proof of the bounded convergence theorem for Riemann integrals. An effort has been made to keep the exposition concise and self-contained.

MSC 2020: 26A42

Acknowledgements

The author would like to thank the University of North Carolina at Charlotte for the continuing support while he was in graduate school and his late aunt Ela Martinović because she cared.

References

[1] C. Arzelà, Sulla integrazione per serie, Atti Acc. Lincei Rend. 4 (1885), 532–537, 596–599. Suche in Google Scholar

[2] N. de Silva, A concise, elementary proof of Arzelà’s bounded convergence theorem, Amer. Math. Monthly 117 (2010), no. 10, 918–920. 10.4169/000298910x523407Suche in Google Scholar

[3] J. J. Duistermaat and J. A. C. Kolk, Multidimensional Real Analysis II, Cambridge Stud. Adv. Math. 86, Cambridge University, Cambridge, 2004. 10.1017/CBO9780511616723Suche in Google Scholar

[4] S. R. Ghorpade and B. V. Limaye, A Course in Calculus and Real Analysis, 2nd ed., Undergrad. Texts Math., Springer, New York, 2018. 10.1007/0-387-36425-0Suche in Google Scholar

[5] R. A. Gordon, A convergence theorem for the Riemann integral, Math. Mag. 73 (2000), no. 2, 141–147. 10.1080/0025570X.2000.11996822Suche in Google Scholar

[6] F. Hausdorff, Beweis eines Satzes von Arzelà, Math. Z. 26 (1927), no. 1, 135–137. 10.1007/BF01475447Suche in Google Scholar

[7] W. J. Kaczor and M. T. Nowak, Problems in Mathematical Analysis. III: Integration, Stud. Math. Libr. 21, American Mathematical Society, Providence, 2003. 10.1090/stml/021Suche in Google Scholar

[8] J. W. Lewin, The teaching of mathematics: A truly elementary approach to the bounded convergence theorem, Amer. Math. Monthly 93 (1986), no. 5, 395–397. 10.1080/00029890.1986.11971838Suche in Google Scholar

[9] W. A. J. Luxemburg, Arzelà’s dominated convergence theorem for the Riemann integral, Amer. Math. Monthly 78 (1971), 970–979. 10.1080/00029890.1971.11992915Suche in Google Scholar

[10] F. Riesz, Über Integration unendlicher Folgen, Jahresber. Dtsch. Math.-Ver. 26 (1918), 274–278. Suche in Google Scholar

[11] B. S. Thomson, The bounded convergence theorem, Amer. Math. Monthly 127 (2020), no. 6, 483–503. 10.1080/00029890.2020.1736470Suche in Google Scholar

Received: 2023-04-19
Revised: 2023-07-27
Accepted: 2023-07-29
Published Online: 2023-08-09
Published in Print: 2024-05-01

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