回複：When I can't your thanks, I check my answer again,my god.....

Let's go back to the original example which is A1=a=2^.5(about 1.4142) and An+1=a^An.  (1)An is increase. (2)An<2

Proof of (2) by induction

A1=a<2

Assume An<2 then An+1=a^An <a^2=2.

Therefore An<2 for all n.

It shows An is bounded by 2.  According the book, an increasing sequence that is bounded from above has a limit.  Why do you think it diverges?

Thanks a lot.