In the book Relativity book by Wolfgang Rindler, he explains that the Galilean and Lorentz transformations exhaust all the possible transformations. And explaining that he says, "Now, either there is, or there is not, an upper bound to the possible speeds of particles. Suppose, first, that there is. Then, mathematically speaking, there must be a least upper bound, which we will call c. This speed c, whether attained or not by actual particles, must be invariant. For suppose some velocity of magnitude c in an inertial frame S corresponds to one of magnitude c' > c in another inertial frame S'. By continuity there will then exist a velocity of magnitude slightly less than c in S (that is, a possible particle velocity) that still corresponds to one of magnitude greater than c in S', a contradiction. Similarly c' < c can be ruled out. So there exists an invariant speed, which is essentially Einstein’s postulate."
Here I do not understand how is he contradicting the assumption by saying that a velocity slightly less than C in S transforms to a velocity greater than C in S', as both are different inertial frames with different speed limits and so one velocity in S can transform to a velocity which is less than c' but may be greater than c.