Proof that the zeta function converges for Re(s)>1

Question

It would be absolutely fantastic if anybody could give me some guidance on the question above. For me (please correct me if I'm wrong), this question boils down to proving that

$\sum^{\infty}_{n=1}\frac{1}{n^{s}}= \prod_{p \, \mathrm{prime}} \frac{1}{1-p^{-s}}$ converges for Re(s)>1 and so, proving that $p^{-s}$ converges for all $s$. How may I formulate the epsilon delta arguments for this?

Thanks a lot

Answer 1

Hint:

• Prove the Zeta function converges absolutely (p-series test)
• Absolute convergence implies convergence

Answer 2

HINT: Cauchy condensation test.