You don't prove a postulate! Obviously, you need some experimental confirmation of it. Which
@Dale has provided a link too.
Fundamentally, however, the point of SR is not about the speed of light. SR is a theory of spacetime and a theory of particle energy-momentum. The simplest and best confirmation of SR is perhaps what happens when you put a particle in a particle accelerator. Particles may be given energy that would classically represent a speed of hundreds or thousands of times the speed of light. That doesn't happen. Instead, they get closer and closer to the speed of light, in agreement with SR energy-momentum.
This is why it's pointless to suspect that modern physics is based on some semi-religious like pronouncements of A. Einstein that no one ever bothered to check! The reality is that all of the 20th Century physics has been based on theories that are either relativistic, quantum mechanical or both. And the whole body of 20th century experimental physics supports these theories.
The future of physics is extending these theories, where required.