Naive question about the speed of light

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I thought that the idea that the speed of light is a constant in every reference frame came from the Michelson-Morely experiment. But a friend of mine was saying that Einstein supposed that if one could catch up to a light beam then it would no longer be moving and would then not be an electromagnetic wave. This would violate Maxwell's equations. Wouldn't this thought experiment suggest that the speed of light is constant?
 
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The point is that according to Maxwell's equations, if you could move at a different speed than c relative to a light beam(as measured by you), then for you, the electromagnetic wave that the light consists of would vanish.

The practical result of this would be that unless you were perfectly at rest with respect to a light source, you wouldn't be able to see it, car radios wouldn't pick up any stations while driving down the road, etc.

Since none of this happens, we conclude that electromagnetic waves are measured as moving at c in every reference frame.
 
Janus said:
The point is that according to Maxwell's equations, if you could move at a different speed than c relative to a light beam(as measured by you), then for you, the electromagnetic wave that the light consists of would vanish.

The practical result of this would be that unless you were perfectly at rest with respect to a light source, you wouldn't be able to see it, car radios wouldn't pick up any stations while driving down the road, etc.

Since none of this happens, we conclude that electromagnetic waves are measured as moving at c in every reference frame.

Ok so why do you need the Michelson Morely experiment?