For every $s > 1$, let $\zeta(s) = \sum_{n=1}^{+\infty} \frac{1}{n^s}$. Bound $\sum_{n=2}^{+\infty} \frac{1}{n^s}$ by two integrals and deduce $\lim_{s \rightarrow +\infty} \zeta(s) = 1$.
For every $s > 1$, let $\zeta(s) = \sum_{n=1}^{+\infty} \frac{1}{n^s}$. Bound $\sum_{n=2}^{+\infty} \frac{1}{n^s}$ by two integrals and deduce $\lim_{s \rightarrow +\infty} \zeta(s) = 1$.