Putnam 1982 B5

For each $x>e^e$ define a sequence $S_x=u_0,u_1,\ldots$ recursively as follows: $u_0=e$ , and for $n\ge0$ , $u_{n+1}=\log_{u_n}x$ . Prove that $S_x$ converges to a number $g(x)$ and that the function $g$ defined in this way is continuous for $x>e^e$ .