Lower Bound for Zeta function on set of natural numbers

By thinking about areas of rectangles under a curve, we get $\begin{aligned} \zeta(n) = 1 + \sum_{k=2}^\infty k^{-n} &\le 1 + \int\limits_1^{\infty} x^{-n} \, dx \\ &= 1+\frac1{n-1}. \end{aligned}$ So we have $1 < \zeta(n) \le 1 + \frac1{n-1},$ so by the squeeze theorem , $\lim\limits_{n\to\infty} \zeta(n) = 1.$ So the greatest lower bound is $\fbox{1}.$