I am reading The Riemann zeta function by H.M. Edwards , $1974$, pg. $52-53$. Here the author denotes by $\rho$ the non trivial zeros of Riemann zeta function
He proves that the following series in $\rho$ converges uniformly in any disc $|s|\leq K$ when the terms $\rho$ and $1-\rho$ are paired $$\sum_{\rho}\frac{1}{s-\rho}$$ For this he observes $$\left|\frac{1}{s-\rho}+\frac{1}{s-(1-\rho)}\right|=\left|\left[\frac{1}{\left(s-\frac{1}{2}\right)-\left(\rho-\frac{1}{2}\right)}\right]+\left[\frac{1}{\left(s-\frac{1}{2}\right)+\left(\rho-\frac{1}{2}\right)}\right]\right| $$ which gives $$ \left|\frac{1}{s-\rho}+\frac{1}{s-(1-\rho)}\right|=\left|\frac{2\left(s-\frac{1}{2}\right)}{\left(s-\frac{1}{2}\right)^2-\left(\rho-\frac{1}{2}\right)^2}\right|$$ Now we have $$\left|\frac{2\left(s-\frac{1}{2}\right)}{\left(s-\frac{1}{2}\right)^2-\left(\rho-\frac{1}{2}\right)^2}\right|\leq \frac{C}{\left|\rho-\frac{1}{2}\right|^2}$$ for all sufficiently large $\rho$ once $K$ is fixed and because $\sum_{\rho}\frac{1}{|\rho-\frac{1}{2}|^2}$ converges.
Question: How do we have the last inequality for all sufficiently large $\rho$? How does it ensure the uniform convergence?
Can someone please clear this doubt. Thank you.