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There are various different situations in mathematics in which, in order to find a fixed point of $f$, we iterate $f$ on an arbitrary starting value, and show that the process must converge to a fixed point.

What is the earliest known instance of this line of reasoning? I'm looking for examples in which the iteration and the convergence to a fixed point was "explicit", in other words, where the mathematician was consciously aware of what they were doing.

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    $\begingroup$ Would you consider "Hero's method" for approximating square roots as meeting your conditions? It amounts to $x_{n+1}:=1/2(x_n+a/x_n)$ in modern terms, and likely goes back to Archimedes, who was presumably aware of what he was doing but not in terms of convergence and fixed points. The recipe was already known to Babylonians. arxiv.org/pdf/1101.0492.pdf Or do you want something more explicitly involving fixed points? $\endgroup$ – Conifold Nov 6 '15 at 19:20
  • $\begingroup$ @Conifold I want something where the mathematician was explicitly aware of the function of which they were finding a fixed point, where the understood the notion of a fixed point as a value whose image under the application of some transformation or rule was itself, and where they proved or at least asserted that iterating the function would lead to such a point. $\endgroup$ – Jack M Nov 6 '15 at 19:39
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It is always difficult, in most cases impossible, to answer the question of the type "who was the first", especially who was the first to discover some trivial thing, which everyone can see once s/he looks at it. By "trivial" in this case I mean that once you start considering iterations of a continuous function, you immediately discover that they can converge only to a fixed point. Thus a more reasonable question would be "who was the first to consider infinite iteration of continuous functions". The earliest instance that I know is Euler's paper

De formulis exponentialibus replicatis. Acta Acad. Petropolitanae 1 (1777), 38–60.

where he considers the existence of the limit $$x^{x^{x^{...}}}$$ This can be easily reformulated as iteration of exponential function. Euler himself says in the paper that he follows the remarkable work of Marquis de Condorcet. But the paper of Condorcet does not address the question of convergence, he just iterates some formal expressions.

But of course I cannot prove that "Euler was the first", and whatever other answer to this question is given, it can never be proved. That's why I think this is a meaningless question, as all questions of this type are.

Remark 1. Of course, Babylonians had an algorithm of extracting the square root, which is equivalent to the Newton method for the equation $x^2-a=0$. But to say that Babylonians were considering "convergence to a fixed point of iterations of a continuous function" would be a ridiculous anachronism.

Euler at least understood the words "function", "limit" and "fixed point" more or less as we understand them now, though not exactly in the same way.

Remark 2. Perhaps it is interesting to mention that Euler found when the limit exists for real $x$. In the subsequent centuries there was a great interest to these problems in complex domain, and complete understanding of possible limits was achieved only in 2009 (J. Hubbard, D. Schleicher and M. Shishikura Journal: J. Amer. Math. Soc. 22 (2009), 77-117).

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