For every $\delta\gt 0$, the partial sums of the Riemann zeta function $\sum_{n=1}^N n^{-s}$ converge uniformly on $S_{1+\delta}=\{s\in\mathbb{C}:\Re (s)\gt 1+\delta\}$. But the convergence is not uniform on the whole of $S_1=\{s\in\mathbb{C}:\Re (s)\gt 1\}$.
Now the fact that $\zeta$ is analytic on $S_{1+\delta}$ for every $\delta\gt 0$ does imply that $\zeta$ is analytic on the whole of $S_1$ – why? Where does this implication come from?