> [!NOTE] Lemma (upper sum is never smaller than lower 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 $P,Q$ be a [[Finite Partition of Closed Real Interval|finite partitions]] of $[a,b].$ Then $L(f,P)\leq U(f,Q)$where $L(f,P)$ denotes the [[Lower Darboux Sum|lower Riemann sum]] of $f$ with respect to $P$ and $U(f,Q)$ denotes the [[Upper Darboux Sum|upper Riemann sum]] of $f$ with respect to $Q.$ **Proof**: Let $R$ be a [[Refinement of Finite Partition of Closed Real Interval|refinement]] of both $P$ and $Q.$ By [[Upper & Lower Darboux Sums of Refinement]], $L(f,P)\leq L(f,R)\leq U(f,R)\leq U(f,Q).$