**Theorem** For every non-

empty set E of

real numbers that is

bounded above, there exists a

unique real number

*sup E* such that

*sup E* is an upper bound for E
- if
*y'* is any upper bound for E, then *y'* >= *sup E*

**Translation** There exists one and only one smallest upper bound for any non-empty set of real numbers.

**Proof** Let y_{1} be a bound for E. Pick a number x_{1} in E. Since y_{1} is an upper bound for E, x_{1} <= y_{1}. Now consider the midpoint, (x_{1}+y_{1})/2. If it's an upper bound for E, call it y_{2}. If it's not an uppper bound, there exists a number in E greater than (x_{1}+y_{1})/2. Let that number be x_{2} and let y_{2} = y_{1}.

We now have x_{2} and y_{2} such that |x_{2}-y_{2}| <= |x_{1}-y_{1}|/2. Repeat the previous steps to get two sequences, {x_{n}} and {y_{n}} with |x_{n}-y_{n}| <= |x_{1}-y_{1}|/2^{n}. These are equivalent Cauchy sequences and therefore converge to the same limit, y.

y is an upper bound for E since x <= y for all x in E.

y is the *least* upper bound of E. Suppose there is another upper bound, y'. x_{j} <= y' since x_{j} is in E and y' is an upper bound of E. However, y = lim_{j->inf.} x_{j}, so y <= y'.

**Translation** Pick a bound for E and pick a number in E. If their midpoint is a bound, let that be your new bound. If their midpoint is not a bound, that's because we have a number in E greater than the midpoint. Let that be the new number in E. Keep doing this until you're bored out of your wits. You have constructed two sequences of numbers that converge to the same point, the least upper bound.

P.S. for

krimson: The

convergence of Cauchy sequences comes from the

density of the

rationals and the fact that a real number is

constructed from an

equivalence class of Cauchy sequences of rational numbers. In other words, the

completeness of the reals. It does

**not** depend on the existence of the

supremum. (It should be noted, I am constructing the real numbers from equivalence classes of Cauchy sequences of rationals, not from

Dedekind cuts.)