Prove that Riemann zeta function $\zeta(z)$ is defined for $z \in \mathbb{C}$ such that $Rez > 1$

Question

Prove that Riemann zeta function $\zeta(z)$ is defined for $z \in \mathbb{C}$ such that $Re(z) > 1$. $$\zeta(z) = \sum_{n=1}^\infty \frac{1}{n^z}$$ For real values bigger than one it's obvious because such series converge. My question is how to prove that it also stands for complex values. Is it correct if we say that a complex function is "made up of" real and imaginary part and if both of them converges then also complex function converge?

Answer 1

That is the correct definition of a complex-valued series converging, yes. But also, absolute convergence implies convergence (for both real and complex series), and $|1/n^z| = 1/n^{\mathop{\text Re} z}$ is very convenient here.