That's a really bad argument, since between any two irrational numbers, there's an infinite number of rational numbers. Ed: For an explicit construction of suc

That's a really bad argument, since between any two irrational numbers, there's an infinite number of rational numbers. Ed: For an explicit construction of such numbers -- Suppose we have a<b, two irrational numbers. Let b-a = d, the difference between them. Then there exists N, an integer, such that 10^(-N) < d. Let b1 be b truncated at the (N+1)th digit past the decimal point. Then b1 is rational (finitely many digits) and b-b1 < d (and b1 < b), so b1 is in (a,b). You can then add back one digit of b at a time, to get b2, b3, etc which form an infinite sequence of rational numbers converging to b from below. (We get infinite unique numbers from the fact b is irrational. Not all the b_n need be distinct.)