power tower sequence

For positive values of $a$, the power tower sequence

$$a,a^a,a^{a^a},a^{a^{a^a}},\mathrm{\dots }$$

is convergent if and only if

$$\frac{1}{e^e}\leqq a\leqq e^{\frac{1}{e}},$$

approximately

$$0.065989\leqq a\leqq 1.444667.$$

The limit of the sequence is the least real root (http://planetmath.org/Equation) of the equation

$$a^x=x.$$

The proof is found in [1].