# Definition(s) > [!NOTE] Definition (Upper Riemann Sum) > Let $[a,b]$ be a [[Closed Real Interval|closed real interval]]. Let $f:[a,b]\to \mathbb{R}$ be a [[Bounded Real Function|bounded]] [[Real Function|real function]]. Let $P=\{ x_{0},x_{1},\dots,x_{n} \}$ be a [[Finite Partition of Closed Real Interval|finite partition]] of $[a,b].$ For $i=1,\dots,n,$ let $M_{i}$ be the [[Supremum of Set of Real Numbers|supremum]] of the [[Image of a set under a function|image]] of $[x_{i-1},x_{i}]$ under $f.$ Then upper Riemann sum of $f$ with respect to $P$ is given by $U(f,P)=\sum_{i=1}^{n} M_{i}(x_{i}-x_{i-1})$ **Remark**: We may write instead $U(f,P)=\sum_{I\in P} |I| \sup_{I} f $if we take $P$ as a set of intervals. # Properties > [!NOTE] Lemma (All upper sums are bigger than or equal to all lower sums) > Suppose $f:[a,b]\to \mathbb{R}$ is bounded and $P$ and $Q$ are partitions of $[a,b].$ Then $L(f,P)\leq U(f,Q).$ ^d7bb7d *Proof.* Choose $R$ to be a [[Finite Partition of Closed Real Interval#^e69641|common refinement]] of both $P$ and $Q.$ Then $L(f,P)\leq L(f,R)\leq U(f,R)\leq U(f,Q).$ # Applications - [[Darboux Integrable Function]].