Let's say I have a particle in a uniformly acclerated motion such that
$$ a_\mu a^\mu = -g^2, \quad g > 0\ {\rm and\ constant} \tag1 $$
I want to prove that any signal coming from a apace-time point $(t > 0, x < t - 1/g)$ cannot ever reach the particle. From Eq. (1) I know that the equations for the particle's motion are
$$ t = \frac{1}{g}\sinh(gs), \quad x = \frac{1}{g}(\cosh(gs) - 1) $$
Where $s$ is the proper time. Then, the trajectory of the particle is:
$$ x_p(t) = \frac{1}{g}\Big(\sqrt{1 + (gt)^2} - 1\Big) $$
So, in order to prove that it cannot be reached by any signal coming from $(t > 0, x)$ that travels a finite time $\Delta t$, I tried proving that the interval was time-like and see if I got the condition $x < t - 1/g$. So,
$$ \int_t^{t + \Delta t}\ dt^2 - \int_x^{x_p(t + \Delta t)} \ dx^2 < 0 $$
And solving the inequation I got that it was fulfilled for,
$$ x > \frac{1}{g}\Big(\sqrt{1 + (g(t + \Delta t))^2} - 1\Big) + \Delta t\\ x < \frac{1}{g}\Big(\sqrt{1 + (g(t + \Delta t))^2} - 1\Big) - \Delta t $$
This has nothing to do with $x < t - 1/g$, so clearly I'm misunderstanding something. How would you do it?
$\bf{COMMENT}$
The answer provided by Eli is an alternative way to solve the problem but it doesn't explain why mine is wrong. So if anyone can say the reason to that, please comment because I'd love to know it