>>11671101

One of the most basic properties of the real numbers is the Archimedean property, part of which states that there are no infinitesimally small real numbers. More exactly, it states that for all real numbers greater than zero, there are natural numbers n such that 1/n is less than the real number.

This is provable from an even more fundamental property of the reals, the least upper bound property. It states that any nonempty set of real numbers which is bounded above has a least upper bound: a number which is >= all elements of the set, and <= all upper bounds of the set.

Let S be the set of infinitesimals in the reals. S is obviously bounded above by 1, because there wouldn't be infinitesimals greater than one. Then, assume by contradiction that S is nonempty. This means that S has a least upper bound I'll call r.

r < 2r, and since r is an upper bound, 2r cannot be in S, so can't be infinitesimal. Thus, we can pick a natural n such that 1/n < 2r.

r/2 < r, and since r is the least upper bound, there must be at least one infinitesimal i > r/2.

However, 1/(4n) < r/2 < i, meaning i is not infinitesimal. By contradiction, S is empty.