Archimedean property

The Archimedean property says the real line contains no infinitely large numbers and no infinitely small positive ones. However huge a real number is, you can step past it by adding 1 to itself enough times; however tiny a positive real is, repeatedly piling up copies eventually overtakes any target.

Precisely: for any real number x there is a natural number n with n greater than x. Equivalently, for any positive real epsilon there is a natural number n with 1 over n less than epsilon. The two forms are the same statement seen from opposite ends, one about getting large, the other about getting small.

For the reals this is a theorem, not an extra axiom: it follows from the least upper bound property. If the naturals were bounded above they would have a supremum s, but then s minus 1 would fail to be an upper bound, giving a natural number above s minus 1 and hence one above s, a contradiction.

The property fails in some exotic ordered fields, such as fields containing genuine infinitesimals used in nonstandard analysis. There an element can be positive yet smaller than every 1 over n. The reals deliberately exclude such elements, which is why ordinary calculus needs no actual infinitesimals.