/sci/ - Science & Math

I understand the first part of the proof in pic related (that alpha^2 can't be less than 2).
But, I absolutely have no idea how the parts where I put the arrows work.
[math]\frac{2\alpha}{n_0} < \alpha^2 - 2[/math], which makes sense (pretty much copy-paste from the first proof).
What I don't understand is the next line. He has only shown that [math]\frac{2\alpha}{n_0} < \alpha^2 - 2[/math], not that it is equal, so shouldn't the next line be:
[math](\alpha - 1/n_0)^2 > \alpha^2 - 2\alpha/n_0 < \alpha^2 - (\alpha^2 - 2) = 2[/math], which would make the entire thing pointless?