> [!NOTE] Lemma (Finer partitions improve upper & lower sums) > 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 $P$ be a [[Finite Partition of Closed Real Interval|finite partition]] of $[a,b]$ and $Q$ a [[Refinement of Finite Partition of Closed Real Interval|refinement]] of $P.$ > >Then $L(f,P)\leq L(f,Q)\leq U(f,Q)\leq U(f,P)$where $L(f,P)$ and $L(f,Q)$ denote the [[Lower Darboux Sum|lower Riemann sums]] of $f$ with respect to $P$ and $Q$ respectively; $U(f,P)$ and $U(f,Q)$ denote the [[Upper Darboux Sum|upper Riemann sums]] of $f$ with respect to $P$ and $Q$ respectively. **Proof**: Suppose $J$ is an interval in $P$ and that $Q$ breaks it into intervals $J_{1},J_{2},\dots,J_{m}.$ Then the [[Infimum of Set of Real Numbers|infimum]] of $f(J)$ will be the most will be at most the infimum on each $J_{j}.$ So the sums $L(f,Q)$ based in the $J_{j}$ have a total as big as the sum in $L(f,P).$