User blog comment:Edwin Shade/In The Pursuit Of Organization/@comment-1605058-20171030102151/@comment-1605058-20171030210222

I mean the actual definition of \(a\uparrow\uparrow\infty\). The usual definition of \(a\uparrow\uparrow n\) is valid only for \(n\) finite, and to ever talk about \(a\uparrow\uparrow\infty\) we have to define it.

If you take the definition of \(a\uparrow\uparrow\infty\) as the limit of \(a\uparrow\uparrow n\), then nothing you have written shows that it converges for \(a\leq e^{1/e}\). At best you have shown that it doesn't converge for other \(a\).

By the way, you mention "evaluating an infinite power tower from top to bottom". There isn't really any top to start from though!