Given That $T(1)=1$ Solve following recurrence function $T(n)=T(\log n)+O(1)$
I know the answer is $\log^* n$ but don't know how to prove it. What I tried: $\log(n)+\log(n-1)+\log(n-2)+...+1 = \log(n*(n-1)*(n-2)*...)=\log n!$ If I am at the right track what to do next?
Thanks.
P.S: This question is same as this but the answer states:
" The guess can be proven somewhat easily using strong induction."
without providing the proof.