[In here,](https://math.stackexchange.com/questions/505367/surprising-identities-equations) I saw that $$x^{x^{x^{x^{x^{x^{x^{.{^{.^{.}}}}}}}}}}$$ exists as a real number (convergent) if and only if $$x\in[e^{-e}, e^\frac{1}{e}].$$ How can I prove this??