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.