uniform convergence of $L(s,\chi)$ for $\Re(s) ≥ 1 + \delta$ " due to absolute convergence for $\Re(s)>1$?

Question

On page 6 of this link, lemma 2.4 shows $L(s,\chi)$ is absolutely convergent for $\Re(s)>1$.

I understand the proof. However, they also add: "The above proof also shows that for any $\delta > 0$, the above series converges absolutely and uniformly for $\Re(s) ≥ 1 + \delta$ "

I do not see why this is the case.