Let $I=[a,b]$ be an interval of $\mathbb R$ and let $f\colon I\to \mathbb{R}$ be a bounded function. For any finite set of points such that , there is a corresponding partition of $I$ .

Let $C(\epsilon)$ be the set of all partitions of $I$ with $\max (x_{i+1}-x_i)<\epsilon$ . Then let $S^{*}(\epsilon)$ be the infimum of the set of upper Riemann sums with each partition in $C(\epsilon)$ , and let $S_{*}(\epsilon)$ be the supremum of the set of lower Riemann sums with each partition in $C(\epsilon)$ . If $\epsilon_1<\epsilon_2$ , then $C(\epsilon_1)\subset C(\epsilon_2)$ , so $S^{*}(\epsilon)$ is decreasing and $S_{*}(\epsilon)$ is increasing. Moreover, $\lvert S^{*}(\epsilon)\rvert$ and $\lvert S_{*}(\epsilon)\rvert$ are bounded by $(b-a)\sup_x \lvert f(x)\rvert$ . Therefore, the limits $S^{*}=\lim_{\epsilon\to 0} S^{*}(\epsilon)$ and $S_{*}=\lim_{\epsilon\to 0} S_{*}(\epsilon)$ exist and are finite. If $S^{*} = S_{*}$ , then $f$ is Riemann-integrable over $I$ , and the Riemann integral of $f$ over $I$ is defined by \begin{equation*} \int_{a}^{b} f(x)dx = S^{*} = S_{*}. \end{equation*}