A well-known problem in elementary real analysis is to show that no uncountable sum of positive real numbers can be finite (this requires a precise formulation, but that would take me too far afield). I believe I saw this problem myself in a textbook as an undergraduate or grad student. If anyone has a reference (if a textbook, edition number and page number would be helpful; I would expect this to appear as a problem in several real/functional analysis texts), I would be quite grateful. Thank you!
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It is probably worth mentioning that this problem appeared on this site several times. See The sum of an uncountable number of positive numbers and some of the posts linked there. – Martin Sleziak Oct 03 '17 at 23:02
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See also this post (which is a reference request, too): Series on Infinite Sets – Martin Sleziak Apr 28 '18 at 12:09
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An equivalent problem appears as Exercise 0.0.1 on page xii of Terence Tao's An Introduction to Measure Theory.
carmichael561
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I believe this question occurs in W. Rudin's "Real and Complex Analysis" as an exercise at the end of the first chapter.
I do give it as an exercise in real analysis, with a "solution"/discussion, e.g., in the first set of examples/discussions at http://www.math.umn.edu/~garrett/m/real/
paul garrett
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