Riemann's Criterion for Integrability, and the Integral Lies between Lower and Upper Sums

In the setting of The Real Line: Standing Notation and Background for Calculus on Intervals, let $a,b\in\mathbb{R}$ with $a<b$, and let $f:[a,b]\to\mathbb{R}$ be bounded: there is $B\in\mathbb{R}$ with $|f(t)|\le B$ for every $t\in[a,b]$. Let partitions of $[a,b]$ be as in Partitions of a Closed Interval, the Set of Partitions, and Refinements (i), let $U(f,P)$ and $L(f,P)$ be the upper sum and the lower sum of $f$ with respect to a partition $P$, and let Riemann integrability on $[a,b]$ and the integral $\int_{a}^{b}f$ be as in The Riemann Integral on a Closed Interval via Lower and Upper Integrals.

(i) $f$ is Riemann integrable on $[a,b]$ if and only if for every real $\varepsilon>0$ there is a partition $P$ of $[a,b]$ with $U(f,P)-L(f,P)<\varepsilon$.

(ii) If $f$ is Riemann integrable on $[a,b]$, then $L(f,P)\le\int_{a}^{b}f\le U(f,P)$ for every partition $P$ of $[a,b]$.