User blog comment:Wythagoras/Proof that TREE(n+1) divided by TREE(n) is strictly increasing/@comment-99.185.0.100-20150917002400/@comment-5529393-20150917020700

Inequalities like these can be very hard to prove, despite being very, very likely to be true based on the huge growth rates of the functions. For example, I don't know if it has been proven that BB(n+1) > BB(n) for all n;  I suspect not, despite BB(n) growing dramatically faster than TREE(n).