Demonstrate that whichever chosen the inertial system, a particle can not have a speed greater than the speed of light C. In other words show that there are no speeds greater than C in nature.

Let $S$ and $S’$ be two different inertial systems, the inertial system $S$ standing and the inertial system $S’$ having a speed $V$ (along the $x$ axis) with respect to system $S$. $V$ is called “transport speed”. Also, let the speed of a particle be $U$ in the system $S$ and $U’$ in the system $S’$.

The Lorentz transformation for the speeds of the same particle between the systems $S$ and $S’$ reads as

$U’ =\frac{U-V}{1-UV/C^2}$, if the system $S’$ is moving away from the system $S$ and

$U’ =\frac{U+V}{1+UV/C^2}$, if the system $S’$ is moving towards the system $S$

The ‘worst case scenario’ for the speed addition is the second case when the speed of the inertial system is adding with the transport speed $V$. There are 4 possible cases

1. Both $(U$ and $V) < C$

$U+V < 2C$, $1+UV/C^2 < 2$ therefore $U’ < 2C/2 =C$

2. $U < C$, $V = C$

$U+V < 2C$, $1+UV/C^2 =1+U/C < 2$, therefore $U’ < 2C/2 = C$

3. $U = C$, $V < C$

$U+V < 2C$, $1+UV/C^2 = 1+V/C < 2$, therefore $U’ < 2C/2 =C$

4. $U =C$, $V = C$

$U+V = 2C$, $1+ UV/C^2 =2$, therefore $U’ =C$

Hence there is no possibility that the speed $U’$ is greater than $C$ (speed of light) in nature.

Reference: Relativistic Velocities