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LUB1 - Arbitrarily Large Integers
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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.