Albert Einstein did not directly prove that the velocity of light is constant in all inertial frames. Instead, he developed the theory of special relativity, which included the postulate that the speed of light in a vacuum is always the same for all observers, regardless of their relative motion. This postulate, known as the "constancy of the speed of light," was a fundamental assumption of special relativity.
Einstein's theory of special relativity was formulated based on two key principles: the principle of relativity and the constancy of the speed of light. The principle of relativity states that the laws of physics are the same in all inertial frames of reference (frames moving at constant velocity relative to each other). In other words, there is no preferred or absolute frame of reference.
By combining the principle of relativity with the constancy of the speed of light, Einstein derived various consequences, including time dilation, length contraction, and the equivalence of mass and energy (E=mc²). These concepts challenged classical notions of space and time and provided a new framework for understanding the behavior of objects moving at high speeds.
Experimental evidence from the late 19th and early 20th centuries, including the famous Michelson-Morley experiment, had already suggested that the speed of light appeared to be constant, regardless of the motion of the source or the observer. Einstein's theory of special relativity provided a consistent mathematical framework to explain and interpret these experimental results.
It is important to note that Einstein's development of special relativity was not based on a direct proof of the constancy of the speed of light but rather on a reimagining of the fundamental principles of physics to reconcile the experimental observations.