Riemann sum convergence theorem ($\Vert {\mathcal{P}}_n\Vert \to 0⟹\sum (f;{\mathcal{P}}_n,{\vec{\xi }}_n)\to {\int }_a^{b}f$)

For a bounded function on a fixed closed interval, Riemann integrability with Riemann integral $I$ is equivalent to convergence of all tagged Riemann sums to $I$ as the mesh of a partition tends to zero. Precisely, for every $\epsilon >0$ there is $\delta >0$ such that every partition of an interval of mesh less than $\delta$, with any tags in the corresponding closed subintervals, has its Riemann sum within $\epsilon$ of $I$.