LUB1 - Arbitrarily Large Integers

For any real number x, there is an integer n such that .   See ita-2-3-11-pg26.

This says that the integers are not bounded from above, because every real number is not an upper bound for the integers.

Now consider the following argument:
(1) B
(2) A implies not B
Therefore not A.

For the sake of argument, we suppose that statements (1) and (2) are true. The second statement, by the definition of material implication, says that we can't have both A and not (not B) = B be true. That is, we can't have both A and B be true statements. But we have supposed B to be true in statement (1). Therefore A is false so that not A is true.

Now let B be the least upper bound property, and let A be the statement that the integers are bounded from above. We will show that A implies not B, which is statement (2) above.

Assume that the integers are bounded from above and let a be any upper bound for the integers. Then . But also , since n+1 is an integer. Then so that is also an upper bound for the integers. But , so that a is not a least upper bound. Since a was any upper bound, it follows that the integers have no least upper bound, and this result is not B - since the integers are not empty. Thus we have not A: The integers are not bounded from above, which was to be proved.