Archimedean property

The Archimedean property of any ordered algebraic structure, such as a linearly ordered group, and in particular of the system of real numbers, is the property of lacking (non-zero) infinitesimals. Such structures that lack infinitesimals are called Archimedean; those that possess infinitesimals are non-Archimedean. A number x would be infinitesimal if the inequality

continues to hold no matter how large the finite cardinal number n of terms in this sum.

The non-existence of nonzero infinitesimal real numbers follows from the least-upper-bound property of the real numbers, as follows. If nonzero infinitesimals exist, then the set of all of them has a least upper bound c. Either c is infinitesimal or it is not. If c is infinitesimal, then so is 2c, but that contradicts the fact that c is an upper bound of the set of all infinitesimals (unless c is 0, so that 2c is no bigger than c). If c is not infinitesimal, then neither is c/2, but that contradicts the fact that among all upper bounds, c is the least (unless c is 0, so that c/2 is no smaller than c).

Archimedes of Syracuse stated that for any two line segments, laying the shorter end-to-end only a finite number of times will always suffice to create a segment exceeding the longer of the two in length. Nonetheless, Archimedes used infinitesimals in mathematical arguments, although he denied that those were finished mathematical proofs.