This paper proves the original and the equivalent ABC conjectures. The hypothesis for the original conjecture is basically the same as the hypothesis for the equivalent conjectures, and this hypothesis states that there exists only finitely many triples (A, B, C) of coprime positive integers, with A + B = C. The conclusion for the original conjecture would be that the product, d, of the distinct prime factors of A, B and C, is usually not much smaller than C. This conclusion would be interpreted as |C – d| < ε, where ε is a positive real number. The conclusions for the equivalent conjectures would be the following: 1. C > rad(d)^(1 + ε), 2. C< K(rad(d))^(1 + ε), where K is a constant, and K is a function of ε, a positive real number, 3. q(a, b, c) = (logC/(log(rad(d)))) > 1 + ε; However, for the equivalent conjectures, one will apply the conclusions containing the constant K, since their solutions for ε can readily be applied in the epsilon-delta proofs in this paper. One will also introduce the constant K into q(a, b, c) = (logC/(log(rad(d)))) > 1 + ε to obtain (logC/(log(rad(d)))) < K(1 + ε). Thus, the conclusions to be used for the equivalent conjectures are 1, C< K(rad(d))^(1 + ε), equivalently, {(logC – logK -log(rad(d)))/(log(rad(d)))} < ε; 2. {(logC/(log(rad(d)))} < K(1 + ε), equivalently, {(logC– Klog(rad(d)))/(Klog(rad(d)))} < ε. Let H = | A + B – C |. Then |H| < δ (δ being a positive real number) would be the hypothesis; and let |L| < ε be the conclusion for the original conjecture with L = C – d. For the equivalent conjectures, let L < ε be the equivalent conclusion, where L = (log C– logK – log(rad(d)))/(log(rad(d))); L = (logC– Klog(rad(d)))/(Klog(rad(d))). It has been proved that if | A + B – C | < δ, then for the original conjecture, |L| < ε; and for the equivalent conjectures, L < ε.