Changes

::''The rational numbers do not satisfy the least upper bound property.''

::''The rational numbers do not satisfy the least upper bound property.''

For example, if we can only use rational numbers, the set of numbers that have squares less then 2 has no rational least uper bound.  1.4142136 is an upper bound, but 1.41421357 is a smaller one.  The exact square root of 2 is the least uper bound that we need, but it isn't rational.

For example, if we can only use rational numbers, the set of numbers that have squares less then 2 has no rational least uper bound.  1.4142136 is an upper bound, but 1.41421357 is a smaller one.  The exact square root of 2 is the least upper bound that we need, but it isn't rational.

==Two ways to define the reals formally==

==Two ways to define the reals formally==