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Every serious source I consulted, be it Cajori, Struik, Edwards,... discusses the method of exhaustion as the means used by ancient Greeks to avoid “taking limits”, because they “disliked infinity”.

As far as my sources go, exhaustion is defined as proving that a sequence of values gets arbitrarily close to some value L by “doing a double reductio ad absurdum to guarantee the sequence does not get arbitrarily close to something less than L, nor something more than L”.

Then they all go on to say that this “avoids the step of taking a limit”. HOWEVER, the definition sounds exactly like what the modern definition of limit is! The only practical difference I see is the consistent use of reductio to reach the result.

What subtlety am I missing?

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  • $\begingroup$ Saying that they used certain methods to avoid taking limits is like saying that they rode horses to avoid getting stuck in traffic on the freeway. $\endgroup$ – Ben Crowell Dec 28 '18 at 20:30
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You are right, their arguments are equivalent to our proofs with limits. An important difference however is the assumption that the limit EXISTS which they took for granted, and which they could not justify because they did not have any definition of real number.

For example, Archimedes says that perimeter of any inscribed polygon in a circle is less than perimeter of any circumscribed polygon. Then he proves that the difference of perimeters of inscribed and circumscribed polygon can be arbitrary small.

One is tempted to conclude that there is a real number which is greater than perimeter of any inscribed polygon and less than perimeter of any circumscribed polygon, and call this number the length of circle. But what is a (real) number, and why such a real number exists?

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