What properties are used to assert that there is always a number between two given numbers?

What properties of "numbers" are used to assert that for given numbers $a$ and $b$, $a≠b$, there exists a number $x$ such that $a < x < b$ ?

In the texts I've read, this seems to be assumed without explanation in discussions that are otherwise quite careful about such things.

For example, in Spivak's Calculus (4E, p. 123) this fact is used to demonstrate that the function $f(x) = x^2$ does not take on its maximum on the interval (0,1) because for any $0 ≤ y < 1$ there is an $x$ such that $y < x < 1$. Up to that point, the only properties of "numbers" (whatever they may be) that have been defined are those of an ordered field, and it is not clear to me that this property can be derived from those.

I gather this amounts to the "numbers" in question having "dense order". (Correct me if I'm wrong.) But I'm unclear where that comes from.

Also note: up to the point where this question arises, there has been no mention of what "numbers" are (i.e., whether they are $\mathbb Q$ or $\mathbb R$), only that they have the properties of an ordered field.

Solution 1:

It is easy to show that in any ordered field if $a < b$, then $a < \frac{a + b}{2} < b$. In any ordered field 2 is invertible, so this intermediate value exists.