Limit of a sequence without using log

For $n=2,3,4,\dots$, define

$$r_n=\min\left\{r\in\mathbb Z\colon\, r\geq 1\,\text{ and }\,\frac{1}{r}+\frac{1}{r+1}+\dots+\frac{1}{n-1}\leq 1\right\}.$$

Consider the sequence$$s_n=\frac{r_n}{n}.$$

One can show by using the approximation $\frac{1}{r}+\frac{1}{r+1}+\dots+\frac{1}{n-1}\sim\ln\left(\frac{n}{r}\right)$ that $s_n\to e^{-1}$ as $n \to\infty$.

Question: Is there a more "direct" way of showing that $s_n=e^{-1}$, say by comparing $s_n$ with $\sum_{k=0}^\infty\frac{1}{k!}$ or $\lim_n\left(1+\frac{1}{n}\right)^n$, and using the root test for example?