> [!NOTE] Theorem > Let $[a,b]$ be a [[Closed Real Interval|closed real interval]]. Let $f,g:[a,b]\to \mathbb{R}$ be [[Darboux Integrable Function|Darboux integrable functions]] such that for all $x\in[a,b],$ $f(x)\leq g(x)$. Then their [[Darboux Integral|Darboux integrals]] satisfy $\int_{a}^{b} f(x) \, dx \leq \int _{a}^{b} g(x) \, dx $ **Proof**: Let $\varepsilon>0.$ By [[Characteristic Property of Infimum of Subset of Real Numbers]], there exists a [[Finite Partition of Closed Real Interval|finite partition]] $P$ of $[a,b]$ so that the [[Upper Darboux Sum|upper Darboux sum]] of $g$ with respect to $P$ satisfies $U(g,P)\leq \overline{\int} g + \varepsilon$Now $U(f,P)\leq U(g,P)$ follows from $f \leq g.$ Thus the [[Upper Darboux Integral|upper Darboux integrals]] satisfy, $\overline{\int} f \leq U(f,P)\leq U(g,P)\leq \overline{\int} g + \varepsilon$that is $\int f \leq \int g $since $\varepsilon>0$ and $\overline{\int} f = \int f$ and $\overline{\int} g = \int g.$