Integrability of $\sin \frac1x$ on $[0,1]$ using Darboux sums

Fix $n$ large. Then for $1/n\le x <y\le 1,$ the mean value theorem shows

$$\tag 1 |f(y)-f(x)|\le n^2|y-x|.$$

Now let $m\in \mathbb N$ be greater than $n^3.$ Set

$$P_n=\{0\} \cup\{1/n+k\frac{(1-1/n)}{m},k=0,1,\dots ,m\}.$$

Using $(1),$ we get

$$U(f,P_n)-L(f,P_n)\le \frac{2}{n} +\sum_{k=1}^{m} \left (n^2\cdot \frac{1-1/n}{m}\right)\cdot \frac{1-1/n}{m}$$ $$ \le \frac{2}{n} + m\frac{n^2}{m^2} < \frac{2}{n} + \frac{1}{n} = \frac{3}{n}.$$