This problem involves pointing out the unjustified inference/logic error in the following proof that all positive rational numbers can be written in "lowest terms" that is as a ratio of positive integers with no common prime factor.
Bogus proof:
Suppose to the contrary that there was positive rational, q, such that q cannot be written in lowest terms. Now let C be the set of such rational numbers that cannot be written in lowest terms. Then q belongs in C, so C is nonempty. So there must be a smallest rational, q0 that belongs in C. So since q0=2 < q0, it must be possible to express q0=2 in lowest terms, namely, q0/2 = m/n, for positive integers m, n with no common prime factor. Now we consider two cases: Case 1: [n is odd]. Then 2m and n also have no common prime factor, and therefore q0 = 2 * (m/n) = (2m/n) expresses q0 in lowest terms, a contradiction.
Case 2: [n is even]. Any common prime factor of m and n/2 would also be a common prime factor of m and n. Therefore m and n/2 have no common prime factor, and so q0 = m / (n/2) expresses q0 in lowest terms, a contradiction.
I've been reasoning through each of the cases and the logic makes sense to me. Any hints or guidance?
Thanks.