User blog comment:Edwin Shade/In The Pursuit Of Organization/@comment-1605058-20171030102151/@comment-1605058-20171031130320

I don't think your argument works even for \(a=e^{1/e}\). It's not because there is a flaw in the proof, it's just because it simply doesn't establish anything at all. You are trying to prove that \(a\uparrow\uparrow\infty\) converges, i.e. that your sequence \(Z_n\) converges. But in your argument you don't even mention \(Z_n\), so it's hard for the argument to establish anything about it!