How does this proof demonstrate the concept of time dilation?

[FactChecker]

They show that the time-dilation is "real" for one of the clocks, and not just symmetric "appearances".

I didn't mean to imply that the time dilation was not real. Both are real and observable by the other coordinate system, but not within it's own coordinate system. That is why the speed of light is the same for all inertial coordinate systems. I was just trying to explain how both A and B could both think that the other's clock was going slower. It is not possible to compare the same clocks side-by-side over any length of time because the clocks move relative to each other. You can only compare separated clocks that have been synchronized in each coordinate system.

I talked about A observing different clocks in B as time progressed, but the argument could be made that A can watch the same clock in B but at different locations in his own coordinate system. The clocks at the different A locations are synchronized in A's coordinate system, and the result is the same.

 

[Markus Hanke]

So B sees A run slow. Please tell me how A can then see B run slow?

Imagine two people standing on a flat plane very far apart. Person A sees person B somewhere in the distance, and perceives him to be very small. Person B looks at person A in the distance, and also perceives him to be very small. The laws of optical perspective are perfectly symmetric in this scenario, and I'm sure no one would argue that there is any kind of paradox arising from this. It's just common sense - both of these observers are right, in their own frames of reference. There is no contradiction.

The same is true for inertial frames - to get from frame A to frame B, you perform a hyperbolic rotation about some angle in space-time ( i.e. a Lorentz transformation ). To get back from frame B to frame A, you perform another rotation about the same angle. And again, there is no mystery or paradox to this - it's a perfectly symmetric scenario, just like the simple thought experiment about the observer in the distance. In many ways, going from one frame to another is really just a change in perspective - choosing a different point of view in space-time, if you will - and one that is identical to its inverse operation.