There’s another sense in which the rational numbers are lacking
and the real numbers fix them up. This one is completely about
the order structure on , and will lead to another construction
of the real numbers. Okay, so what’s wrong with the rational
numbers now? In any partial order we can consider least upper or
greatest lower bounds. That is, given a nonempty set of rational
numbers the least upper bound or “supremum” is a rational number
so that for all — it’s an upper bound — and if is any upper
bound then . Similarly the “infimum” is the greatest lower bound
— a lower bound that’s greater than all the other lower bounds.
There’s just one problem: there might not be a supremum of a
given set. Even if the set is bounded above, there may be no
least upper bound. For instance, the set of all rational numbers
so that is bounded above, since is an example of an upper bound,
and is another. But no matter what upper bound we have in hand,
we can always find a lower number which is still an upper bound
for this