There is a standard trick in analysis, where one chooses a subsequence, then a subsequence of that... and wants to get an eventual subsubsequence of all of them and you take the diagonal. I've always called this the diagonalization trick.

I heard once that this is due to Cantor but haven't been able to find a reference (all searches for diagonal and Cantor lead to his argument about the uncountability of [0,1].

Does anyone have an exact reference?



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    $\begingroup$ Reed & Simon here ;-) $\endgroup$ – Francois Ziegler Apr 26 '14 at 1:54
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    $\begingroup$ I wouldn't be surprised if it's only the idea of diagonalization that is due to Cantor (in his proof of uncountability of $[0,1]$) and its particular manifestation for iterated subsequences was due to someone else later. $\endgroup$ – KConrad Apr 26 '14 at 2:04
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    $\begingroup$ This isn't really a "diagonalization argument" in the modern sense used by logicians, though it does share some key ideas. Rosenthal has called this type of argument "Ramsey's Theorem for Analysts", see my answer here, so it could as well be called a "Ramsey trick". $\endgroup$ – François G. Dorais May 4 '14 at 14:01

From this, it sounds like a very early instance is in Ascoli's proof of his theorem: pp. 545-549 of Le curve limite di una varietà data di curve, Atti Accad. Lincei 18 (1884) 521-586. (Which, alas, I can't find online.)

Note that this predates Cantor's argument that you mention (for uncountability of [0,1]) by 7 years.

Edit: I have since found the above-cited article of Ascoli, here. And I must say that the modern diagonal argument is less "obviously there" on pp. 545-549 than Moore made it sound. The notation is different and the crucial subscripts rather hard to read, so at first sight I feel the need for a native Italian speaker to help spot where precisely Ascoli passes to the diagonal sequence (as I guess he must somehow)...

(One may also consult a self-summary of the article in Rend. Reale Istituto lombardo di scienze e lettere 21 (1888) 365-371 here.)

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    $\begingroup$ You may want to see this as well. Du Bois-Reymond's paper is from 1875, and it is a very similar diagonalization argument to the "trick" the question refers to. $\endgroup$ – Andrés E. Caicedo Apr 26 '14 at 18:41
  • $\begingroup$ @AndresCaicedo Thanks, I guess you mean the footnote on p. 365 of this long paper. You made me realize that there is a whole literature on this; e.g. the question whether du Bois-Reymond anticipates Cantor (or perhaps rather Ascoli) is interestingly debated here and here. $\endgroup$ – Francois Ziegler Apr 28 '14 at 1:56
  • $\begingroup$ @FrancoisZiegler, it is one of the most stunning pieces of argument out there, so it is rather natural that its genesis attracts a lot of interest. $\endgroup$ – Mariano Suárez-Álvarez May 3 '14 at 5:56

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