Proving parallel rays converge to focus using Fermat's principle for a concave mirror

Fermat’s principle is a boundary value problem which gives possible paths as it’s answer. And I understand how it’s applied to give possible paths between two arbitrary points. And following from the definition of a parabolic surface it could be easily shown that the focus point can only receive rays that are parallel to the principal axis from any arbitrary point P. This follows from the fact that any other path (not parallel) would not be the one with least time (Fermat’s principle and geometrical definition of a parabola). That just implies that if a ray were to pass through the focus it should be parallel. But in order to prove that ANY RAY IF THEY ARE PARALLEL WILL CONVERGE to the focus, it has to be shown that the parallel rays that do converge to the focus exhausts ALL POSSIBLE parallel rays that exists such that it does not LEAVE any single parallel ray to converge on any another point (other than the focus) after reflection. That is we have to negate a situation wherein a ray is parallel to the principle axis but after reflection falls on a point other than the focus. And we can do that by the following argument : If a parallel ray from point P falls on a point S’ other than the focus S after reflection, then there exists another ray from P that is not parallel (since the parallel ray falls into S’ and not S) that converges into S thereby contradicting the conclusion that the focus only accepts rays that are parallel to the principal axis. But that requires you to prove that given any two arbitrary points P (on the plane) and S’ (on the principal axis) THERE HAS TO EXIST A PATH such that a ray from P following that path DOES reach S’. Only then can you say that if a parallel ray from a point P does not fall on the focus there has to be some other path from the same point P that does fall on the focus but since that would not be parallel, it violates the conclusion that focus accept parallel rays only.

But proving that requires context of what kind of reflecting surface is being considered and in none of the textbooks I read have I seen this requirement. Is there a mistake in my argument because I am missing something basic or is it assumed without mentioning that given any two points there exists at least one path (got by adjusting the angle the ray makes with the principal axis) that connects them.