...greater than the half, if from the remainder [a part] greater than the half be subtracted, and so on continually, there will be left some magnitude which will be less than the lesser given magnitude." This last lemma is frequently assumed by Archimedes, and the application of it to...

...greater than the half, if from the remainder [a part] greater than the half be subtracted, and so on continually, there will be left some magnitude which will be less than the lesser given magnitude." This last lemma is frequently assumed by Archimedes, and the application of it to...

...half, and from that which is left a magnitude greater than its half, and if this process be repeated continually, there will be left some magnitude which will be less than the lesser magnitude set out." This principle is deduced by Euclid from the axiom that, if there are two magnitudes of the same kind,...

...there be subtracted more than its half (or the half itself), from the remainder more than its half (or the half), and if this be done continually, there...some magnitude which will be less than the lesser of the given magnitudes. This last lemma is frequently used by Archimedes himself (notably in the second...

...there be subtracted more than its half (or the half itself), from the remainder more than its half (or the half), and if this be done continually, there will be left some niagmtude which will be less than the lesser of the given magnitudes. This last lemma is frequently...

...half, and from that which is left a magnitude greater than its half, and if this process be repeated continually, there will be left some magnitude which will be less than the lesser magnitude set out. . This result, which we will call "Eudoxus' principle," may be phrased as follows. Let A/o and e be...

...and from the greater a magnitude greater than its half is subtracted, and this process is repeated continually, there will be left some magnitude which will be less than the lesser magnitude set out. This means that any magnitude is infinitely divisible; that there is no smallest magnitude, because...

...which is left a magnitude greater than its half, and if this process be repeated continually, then there will be left some magnitude which will be less than the lesser magnitude set out. At the conclusion of the proof Euclid says the theorem can be proven if the parts subtracted be halves....

...half, and from that which is left a magnitude greater than its half, and if this process be repeated continually, there will be left some magnitude which will be less than the lesser magnitude set out. Application of this principle shows that, for any e > 0, we obtain after a finite number of steps an...

...half, and from that which is left a magnitude greater than its half, and if this process be repeated continually, there will be left some magnitude which will be less than the lesser magnitude set out. (Elements X, 1) The general procedure for an exhaustion proof is to begin with upper and lower bounds...