Originally Posted by John Henry

Why didn't I think of looking that up before? A quick Google turned up multiple proofs of this, including one that generalised to other square roots.

Clearly, then, irrational numbers do exist. Does any more general or generalisable proof exist that all non-terminating, non repeating real numbers are irrational?

I don't entirely understand the assertion that the set of integers contains no infinite numbers, which doesn't mean I don't believe it, but my problem is as follows:

For any finite set of consecutive, positive integers starting with 1, the value of the largest member is equal to the magnitude of the set.

ie The set |A| is the first n consecutive integers greater than zero.

|A|=n
A={1,2,3...n}

The largest element of this set is always equal to n.

eg

n=9
|A|=9
A={1,2,3,4,5,6,7,8,9}

n=10
|A|=10
A={1,2,3,4,5,6,7,8,9,10}

n=1000
|A|=1000
A={1,2,3...1000}

The set of all integers is infinite we can set n as infinite. If we set n infinite, then we have a set of integers which contains n, which is infinite, giving us an infinite integer.

...don't we?