An elementary proof of the convergence of iterated exponentials (Q1349447)

The author proves the following known result: For a fixed positive real number \(a\) the sequence \(a_1:=a\), \(a_{n+1}:=a^{a_n}\) \((n\geq 1)\) converges if and only if \(e^{-e}\leq a\leq e^{{1\over e}}\). (Reviewer's remarks: A large number of references on this topic was collected by \textit{R. A. Knoebel} [Am. Math. Mon. 88, 235-252 (1981; Zbl 0493.26007)]. For more recent results see \textit{G. Bachman} [Pac. J. Math. 169, No. 2, 219-233 (1995)] and \textit{I. N. Baker} and \textit{P. J. Rippon} [Complex Variables, Theory Appl. 12, No. 1-4, 181-200 (1989; Zbl 0644.30014)] and the references given there).