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.
 A: I think what he is saying is simply that if there exists a speed limit then the transformation law for velocity must be of such a nature that the speed that corresponds to the speed limit must remain invariant under such transformations. Because otherwise the law of the speed limit itself will be violated. 
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.
A: Alright, so I was strugling with this exact page and I think I can give an explanation. So, the essential point to be shown is that there must always exist some sort of speed limit (whether that be finite or infinity). By the switching frame trick he does, he concludes that if you find a least upper bound, then you can always find a bound less than that which is again a least upper bound.
This means that there is no LUB. If you want analogous situation, I suggest you check out Rudin's Real Analysis book and how he shows the upper bound of numbers getting close to $\sqrt{2}$.
