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The proof relies on the fact that each partial sum $f_n(s) = \sum_{k=1}^n \frac{1}{k^s}$ is a finite sum of analytic functions, making $f_n(s)$ itself analytic. The lecture previously established that $f_n(s)$ converges uniformly to $\zeta(s)$ on any closed half-plane $\operatorname{Re}(s)\ge 1+a$ for $a>0$.

Conditions: $f_n(s)$ are the partial sums of the zeta series.; The Uniform Limit Theorem applies to sequences of analytic functions converging uniformly on a domain.; $a$ is an arbitrarily small positive constant ($a>0$).

The proof relies on the fact that each partial sum $f_n(s) = \sum_{k=1}^n \frac{1}{k^s}$ is a finite sum of analytic functions, making $f_n(s)$ itself analytic. The lecture previously established that $f_n(s)$ converges uniformly to $\zeta(s)$ on any closed half-plane $\operatorname{Re}(s)\ge 1+a$ for $a>0$.

Conditions: $f_n(s)$ are the partial sums of the zeta series.; The Uniform Limit Theorem applies to sequences of analytic functions converging uniformly on a domain.; $a$ is an arbitrarily small positive constant ($a>0$).