User blog comment:Ikosarakt1/Fast-growing hierarchy/@comment-5529393-20130627000838/@comment-5529393-20130628143233

It is true that \(\Omega^2 + \Omega\) is the limit of \(\Omega^2 + \alpha\) for countable \(\alpha\), but that doesn't affect the fact that  \((\Omega^2 + \Omega\)*\Omega\) ls the limit of \((\Omega^2 + \Omega\)*\alpha\) for countable \(\alpha\), and that limit is \(\Omega^3\), not \(\Omega^3 + \Omega^2\), as one can verify that \((\Omega^2 + \Omega\)*\alpha < \Omega^3\) for countable \(\alpha\).

It is not true in general that (a+b)*c = a*c + b*c, even for countable a,b,c, as for example \((\omega^2  + \omega) * \omega) = \omega^3\), since \((\omega^2  + \omega) * \omega)\) is the limit of \((\omega^2  + \omega) * n\) for finite n, and \((\omega^2  + \omega) * n < \omega^3\) for finite n.

It is true in general that a * (b+c) = a*b + a*c.