Proposition 4

If (x, y) is any pair of real numbers such that x < y, there is a rational number r such that x < r < y.

Since \mathbf{Q} is dense in \mathbf{R}, it is enough to show that ]x, y[ is not empty; by translation we may assume x = 0 and y > 0. Now \mathbf{R} is a Hausdorff space and therefore, by Proposition 3, there is a rational number r > 0 such that y \notin ]-r, +r[, and this implies that 0 < r < y.