Firstly this only works when n>4, so you can't start with n=1 (whilst it does work for 1, it wont for 2, 3, 4)
Theres a trick in here.
You have shown that 2<sup>k</sup> > k<sup>2</sup>
Which means that 2.2<sup>k</sup> > 2.k<sup>2</sup> (Just multiplied by 2 on each side)
So when you...