Solving Infinite Power Tower Equation: Can we find a solution?

This video presents an infinite power tower equation, also known as a hyper-4 equation or exponentiation tower, and examines whether a solution can be found. This is also sometimes referred to as Tetration or SuperExponentiation or infinite left exponent. Repeated exponentiation in infinite power towers has a limited convergence range, so it is important to exercise caution when attempting to solve them. In general, Euler proved that the infinitely iterated exponential converges for exp(−e) ≤ x ≤ exp(1/e) that translates to approximately the interval from 0.066 to 1.44 for x. This interval includes the choice of x=sqrt(2).

[erikev]

Could you convince me that this is sound math? How can you prove the limit converge? If it converges, how can you prove the solution unique?