Why do principal energy levels in an atom get closer together as n increases? The title says it all. Reasons that I can supply include: increased nuclear charge increasingly catches up in terms of influence to the increasing shielding and proof by contradiction in that if the space remained the same there would be negligible nuclear pull on the valence electrons and so this must not be the case. What is a comprehensive explanation for why this phenomenon is so?

Thank you!
The energy levels for an atom are similar to a "particle in a box", except that the walls have a different shape. When you analyzed the "particle in a box" situation, you used a rectangular box. An atom is spherical, so the walls of the atom open outwards at the same rate as the sphere increases in surface area. You should have noticed that the energy levels for a "particle in a box" decreased in distance at higher energy levels. The surface area of the sphere brings the energy levels even closer.