Darboux Sums Bound the Lebesgue Integral on a Closed Interval

In the settings of Measurable Functions and the Lebesgue Integral: Notation and The Real Line: Standing Notation and Background for Calculus on Intervals, assume the axiom of dependent choice. Let $a,b\in\mathbb{R}$ with $a<b$, let $[a,b]$ be the closed interval, and let $\lambda$ be Lebesgue measure on $\mathbb{R}$. Write $([a,b],\mathcal{B}_{[a,b]},\lambda_{[a,b]})$ for the restriction of $(\mathbb{R},\mathcal{B}(\mathbb{R}),\lambda)$ to $[a,b]$ (a measure space, as $\lambda$ is a measure and $[a,b]\in\mathcal{B}(\mathbb{R})$ by Lebesgue Measure on the Real Line: Existence, Lengths of Intervals and Uniqueness (iv) and Lebesgue Measure on the Real Line: Existence, Lengths of Intervals and Uniqueness (i)); thus $\mathcal{B}_{[a,b]}=\{D\in\mathcal{B}(\mathbb{R}):D\subseteq[a,b]\}$ and $\lambda_{[a,b]}(D)=\lambda(D)$ for $D\in\mathcal{B}_{[a,b]}$. Let $f:[a,b]\to\mathbb{R}$ be measurable with respect to $\mathcal{B}_{[a,b]}$ and bounded: there is $B\in\mathbb{R}$ with $|f(t)|\le B$ for every $t\in[a,b]$. For a partition $P$ of $[a,b]$, $L(f,P)$ and $U(f,P)$ are the lower sum and the upper sum of $f$ with respect to $P$.

(i) Darboux bounds. $f\in\mathcal{L}^{1}(\lambda_{[a,b]})$, and for every partition $P$ of $[a,b]$,

$$L(f,P)\le\int f\,d\lambda_{[a,b]}\le U(f,P).$$