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 2*c*, but that contradicts the fact that *c* is an upper bound of the set of all infinitesimals (unless *c* is 0, so that 2*c* 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.