Can someone prove that all parallel lights beams reflect back and meet at point the point $F$? Any help would be appreciated.
Strictly speaking, it doesn't (in general) in case of spherical mirrors. This is a very important property of parabolas that any number of parallel lines (parallel to the axis of the parabola) reflect and converge at the focus.
As you learn more optics you'll be familiar with the fact that as you make the spherical lens smaller and smaller it approaches a parabola so we can treat it like one. Hence this approximation works. It won't work with bigger spherical lenses as the curvature of the lens will be very different from that of a parabola.
More on parabolas here.
Happy learning :-)