Numbers

4 Least upper bounds

4.3 Least Upper Bound Property

In the examples just given, it was straightforward to guess the values of sup E and inf E. Sometimes, however, this is not the case. For example, if

then it can be shown that E is bounded above by 3, but it is not so easy to guess the least upper bound of E.

In such cases, it is reassuring to know that sup E does exist, even though it may be difficult to find. This existence is guaranteed by the following fundamental result, on which many other results in analysis are based.

(The Least Upper Bound Property of is an example of an existence theorem, one which asserts that a mathematical object, such as a real number, exists with a certain property.)

The Least Upper Bound Property of is very plausible geometrically. If the set E lies entirely to the left of some number M, then you can imagine decreasing the value of M steadily until any further decrease causes M to be less than some point of E. At this point, sup E has been reached.

The Least Upper Bound Property of can be used to show that includes decimals which represent irrational numbers such as , as claimed earlier, and also to define arithmetic operations with decimals. (See Section 5.2.)

There is a corresponding property for lower bounds.

Finally we prove the Least Upper Bound Property in the case when the set E contains at least one positive number. The proof in the general case can be reduced to this special case; we omit the details.

Proof We know that E is bounded above and contains at least one positive number. We can now apply the following procedure to give us the successive digits in a decimal a₀.a₁a₂ …, which we then prove to be the least upper bound of E.

We choose in succession:

the greatest integer a₀ such that a₀ is not an upper bound of E;

the greatest digit a₁ such that a₀.a₁ is not an upper bound of E;

the greatest digit a₂ such that a₀.a₁a₂ is not an upper bound of E;

·

·

·

the greatest digit a_n such that a₀.a₁ … a_n is not an upper bound of E;

Thus at the nth stage we choose the digit a_n so that

We now use Strategy 4.1 to prove that the least upper bound of E is

First we have to show that a is an upper bound of E; that is, if x E, then x ≤ a. To do this, we prove the contrapositive statement: if x > a, then x E. Let x > a and represent x as a non-terminating decimal x = x₀.x₁x₂.... (For example, if x = 3.2, then we write x = 3.1999....) Since x > a, there is an integer n such that

Hence

so x₀.x₁x₂ … x_n is an upper bound of E, by statement (4.3). Since x is non-terminating, x > x₀.x₁x₂ … x_n, so x E, as required.

Next we have to show that if x < a, then x is not an upper bound of E. Let x < a. Then there is an integer n such that

so x is not an upper bound of E, by statement (4.2).

Thus we have proved that a is the least upper bound of E.