User blog comment:LittlePeng9/FORMAL comparison of Hydra battles and Hardy hierarchy/@comment-10429372-20141017061807/@comment-1605058-20141017203739

We can show that, due to how fundamental sequences in Hardy hierarchy work, if we evaluate $$H_{\alpha[n+1]}(a)$$ we will, at some point, hit the expression of the form $$H_{\alpha[n]+1}(b)$$ for $$b\geq a$$. If anyone is interested, I can think about details of the proof tomorrow.