The author proves the original ABC conjecture which states that if A, B and C are three coprime positive integers such that A + B = C, and d is the product of the distinct prime factors of A, B and C, then d is usually not much smaller than C. The author will adhere to the wording of the original conjecture and not to any equivalent conjecture, since if one proves an equivalent conjecture, logically, one would also have to prove the equivalency, otherwise, the proof of the original conjecture would be incomplete. The author assumes that the statement " d is usually not much smaller than C" means the difference between C and d is usually less than a small positive number, say, ε.. Then, one obtains |C– d |< ε., which would be the conclusion. If A + B - C = 0, then for a very small positive number, δ > 0, one can write |A + B – C| < δ, From above, the hypothesis would be |A + B – C| < δ, and the conclusion would be |C– d |< ε. The author has proved that if |A + B – C| < δ, then |C– d |< ε.