Continuity of the Riemann zeta function on $\operatorname{Re} z>1$

Show that the series

\[ \sum _ { n = 1 } ^ { \infty } { \frac { 1 } { n ^ { z } } } \]

defines a continuous function for \(\mathrm { R e } ( z ) > 1\) . (This is the Riemann ζ function and you will see later that it is analytic in the above region-not just being continuous thereand it can be extended to the complex plane with only 1 removed. You do not need to prove the latter assertions.) [Hint: Show that the series converges uniformly for \(\mathrm { R e } ( z ) \geq 1 + \delta\) , where \(\delta > 0\) is any positive number.]