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.
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. Search 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. 10.4169/000298910x523407Search 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/CBO9780511616723Search 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. 10.1007/0-387-36425-0Search 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.2000.11996822Search in Google Scholar
[6] F. Hausdorff, Beweis eines Satzes von Arzelà, Math. Z. 26 (1927), no. 1, 135–137. 10.1007/BF01475447Search 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/021Search 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. 10.1080/00029890.1986.11971838Search in Google Scholar
[9] W. A. J. Luxemburg, Arzelà’s dominated convergence theorem for the Riemann integral, Amer. Math. Monthly 78 (1971), 970–979. 10.1080/00029890.1971.11992915Search in Google Scholar
[10] F. Riesz, Über Integration unendlicher Folgen, Jahresber. Dtsch. Math.-Ver. 26 (1918), 274–278. Search in Google Scholar
[11] B. S. Thomson, The bounded convergence theorem, Amer. Math. Monthly 127 (2020), no. 6, 483–503. 10.1080/00029890.2020.1736470Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- Pseudo-n-multipliers and pseudo-n-Jordan multipliers
- Fractional dual Simpson-type inequalities for differentiable s-convex functions
- Fredholm weighted composition operators between weighted lp spaces: A simple process point of view
- The exterior differential operator on quasi-Kähler manifolds and some relations of its components for smooth functions
- Statistical approximation using wavelets Kantorovich (p,q)-Baskakov operators
- Arzelà’s bounded convergence theorem
- On the Cauchy problem for the generalized double dispersion equation with logarithmic nonlinearity