LUB5
Arbitrarily Close
Approximation
to Real Numbers
by Rational Numbers

If   x, ε ∈ R,   ε > 0,
then there exists
a rational number r
such that
| 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/Nx < (n+1)/N .   Then   0 ≤ x - n/N < 1/N < ε,   so   | x - n/N | < ε .

See ita-2-3-15-pg26.