Invariant upper bound for velocity of a particle

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.

• I have no idea what an invariant that is not actually attained by particles is supposed to mean. Such a thing may have mathematical sense, it sure doesn't have any physical meaning. Jun 14 '16 at 6:23
• In the special theory of relativity, speed of light is the limit to the velocity that can be attained by any particle. Here this upper bound is the same in all reference frames. Now is there a transformation where this upper bound is different for different reference frames. The invariant velocity is being referred to this upper bound of velocity. Jun 14 '16 at 6:26
• @CuriousOne for massive particles, $c$ is precisely such an invariant: $v < c$, not $v \le c$: it's the supremum of velocity, not the maximum.
– user107153
Jun 14 '16 at 6:30
• @tfb: Such a thing is not physically measurable and I am simply going by what can be measured, not what works for a mathematician. Jun 14 '16 at 7:37

As he has said, suppose the rule is that nothing can travel faster than $k$ in any frame. Now if the transformation law for velocity is of such a nature that it transforms the velocity $k$ to a velocity $k'>k$ in another frame then because of continuity there will be particles with speed $w<k$ which can transform to some speed $w'<k'$ but $w'>k$ in another frame. This would violate the law of the speed limit.
The speed limit which is not achievable by particles might be thought of as an asymptotic limit of the speed that particles might attain. We can go as close as it pleases us but can't reach it. In such a case, a plain statement of the transformation of $k$ to $k'>k$ would not as such violate the law of the speed limit as nothing actually travels at that speed. But this would immediately imply the violation of the speed limit if one invokes the continuity argument as he has used.
• The argument is quite physical. It has incorporated the principle of relativity in it. The speed limit it assumes is universal - i.e. for every frame. If the speed limit is $k$ then it doesn't really refer to any particular frame but rather it refers to all the frames. So $w'>k$ in the $O'$ frame violates the speed limit law in $O'$ frame. Jun 14 '16 at 7:05