I am studying about upper bounds for $N(\sigma, T)$ (zero-density estimates), and while going through Theorem 9.16 in Titchmarsh's book (2nd edition) (page 231), I got a bit stuck in understanding the following string of inequalities: $$\Bigg(\sum_{n \geq X} \frac{d(n)}{n^2}\Bigg)^2 = O(X^{2\epsilon -2}) < \frac{1}{2X} < \frac{1}{2}.$$
The last inequality is obvious, but I am kind of stuck understanding the first equality (estimate) and the inequality following that. I kind of sense that I have to make use the fact that $d(n) \ll_{\epsilon} n^{\epsilon}$ somehow, along with the fact that $n \geq X$, but I don't seem to get it. Any help and/or hints is much appreciated. Thanks in advance.