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I am trying to show that timelike geodesics reach the Rindler horizon (X=0) in a finite proper time.

The spacetime line element is

[itex]ds^{2} = -\frac{g^{2}}{c^{2}}X^{2}dT^{2}+dX^{2}+dY^{2}+dZ^{2}[/itex]

Ive found something helpful here:

https://www.physicsforums.com/showpost.php?p=3316839&postcount=375

But don't understand why you have to take T=∞ in order to find X=0?

I also don't understand how to work out that the worldine is given by r=(t,sech(t),0,0).

Any help will be greatly appreciated.