For \(s\in\CC\) with \(\operatorname{Re}s>1\), the Riemann zeta function is \[ \zeta(s)\coloneqq\sum_{n=1}^{\infty}\frac{1}{n^{s}}, \] where \(n^{s}\coloneqq e^{s\log n}\) with the real logarithm. The series converges absolutely because \(\abs{n^{-s}}=n^{-\operatorname{Re}s}\).
(Euler product)
For \(\operatorname{Re}s>1\), \[ \zeta(s)=\prod_{p\text{ prime}}\frac{1}{1-p^{-s}}, \] where the product is over all prime numbers \(p\) and converges as the limit of its partial products over \(p\le N\).
Let \(\sigma\coloneqq\operatorname{Re}s>1\). For each prime \(p\), \(\abs{p^{-s}}=p^{-\sigma}<1\), so \(\frac{1}{1-p^{-s}}=\sum_{k\ge0}p^{-ks}\), an absolutely convergent series. Multiplying these finitely many series for the primes \(p\le N\) and using unique factorization in \(\ZZ\), \[ \prod_{p\le N}\frac{1}{1-p^{-s}}=\sum_{n\in S_N}n^{-s}, \] where \(S_N\) is the set of positive integers all of whose prime factors are at most \(N\). Every \(n\le N\) lies in \(S_N\), so \[ \abs{\zeta(s)-\prod_{p\le N}\frac{1}{1-p^{-s}}}\le\sum_{n>N}n^{-\sigma}\to0\qquad(N\to\infty). \]