User blog comment:B1mb0w/Fundamental Sequences/@comment-26428969-20151104104848/@comment-1605058-20151107081620

This can't be correct. $$(\omega\uparrow\uparrow\beta)^n$$ does not converge to  $$\omega\uparrow\uparrow(\beta+1)$$. Take a look at a small example $$\beta=2$$:  $$(\omega\uparrow\uparrow\beta)^n= (\omega\uparrow\uparrow2)^n=(\omega^\omega)^n=\omega^{\omega n}$$, which clearly converges to $$\omega^{\omega^2}$$, not  $$\omega\uparrow\uparrow3= \omega^{\omega^\omega} $$.