If x, ε ∈ R, ε > 0,
then there exists
a rational number r
| x - r | < ε . (
In other words, a real number may be approximated as closely as we wish by a rational number.
To prove this, use LUB2 to find a positive integer N such that 1/N < ε, then use LUB4 to find an integer n such that n/N ≤ x < (n+1)/N . Then 0 ≤ x - n/N < 1/N < ε, so | x - n/N | < ε .