I understand the proof, through and through. However, I have a logical dilemma involving the proofs conclusion.
The proof doesn't seem valid to me. I looked around the web and saw very similar
descriptions for this proof and they too all seem invalid to me. All the proofs start with the idea that a rational number can be written as the ratio of two integers, say a/b, and that for any ratio there exists exactly one fully reduced fraction (where no integer greater than 1 exists that can be evenly divided into both the numerator and denominator.)
What I see is there may exist a fraction that is not fully reduced. Even if I assumed a non-fully reduced fraction did exist, this does not imply to me there does not exist a non-fully reduced fraction.
In other words, just because you found a solution that proves that the solution itself is not in reduced/lowest terms; it doesn't mean that the solution couldn't possibly be not in reduced lowest terms? I also realized that we assumed that it was in reduced lowest terms.
However, that's like assuming that X is a natural number and 2x + 5 = 6. So you proved that it isn't a natural number, contradicting your assumption - so what?
Every non-fully reduced fraction can be reduce to fully reduced fraction!