Zeta bounds from rounded Pareto variables ($1/(s-1)\le \zeta (s)\le s/(s-1)$)

Let $X$ have survival function $\mathbb{P}(X\ge x)=x^{-s}$ for $x\ge 1$, with $s>1$. Its expected value is $s/(s-1)$. The tail-sum formula for expectation givesThe absolutely continuous probability distribution gives zero mass to the integers, so $\lceil X\rceil =\lfloor X\rfloor +1$ almost surely and $\mathbb{E}[\lceil X\rceil ]=\zeta (s)+1$. Taking expectations of $\lfloor X\rfloor \le X\le \lceil X\rceil$ gives $\zeta (s)\le s/(s-1)\le \zeta (s)+1$, which rearranges to the displayed bounds on the Riemann zeta function. Finiteness is justified by $\lfloor X\rfloor \le X$ and $\lceil X\rceil \le X+1$.