= Riemann sum convergence theorem
{c}
{title2=$\|\mathcal P_n\|\to0\Longrightarrow\sum(f;\mathcal P_n,\vec\xi_n)\to\int_a^bf$}
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 $\varepsilon>0$ 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 $\varepsilon$ of $I$.
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