User blog comment:Wythagoras/Proof that TREE(n+1) divided by TREE(n) is strictly increasing/@comment-223.255.230.227-20141220074940/@comment-1605058-20141220093845

First, ceil(log_3 TREE(3))-1.0000001 is smaller than ceil(log_3 TREE(3))-1. Second, it doesn't necessarily have to fail - if log_3 TREE(3) is of the form k+0.6, for k integer, then ceil(log_3 TREE(3))-1=k, and we have inequality holding even for ceil(log_3 TREE(3))-1+c for c=0.5.