What is the proof of the theorem that the series of reciprocals of square-free numbers diverges?

Question

I am not a mathematician. I am reading David Applebaum's book "Limits, limits everywhere ", in which he gives a proof of the divergence of the series of square-free numbers on page 91. I don't understand where he gets the inequality that he uses to prove the theorem. I also don't understand whether the three sums in the inequality run over the same limits. Any light shed on this issue will be greatly appreciated.