Rules for the Riemann Integral: Sums, Scalar Multiples, Monotonicity, Bounds and Constants

In the setting of The Real Line: Standing Notation and Background for Calculus on Intervals, let $a,b\in\mathbb{R}$ with $a<b$, let integrability on $[a,b]$ and the integral $\int_{a}^{b}$ be as in The Riemann Integral on a Closed Interval via Lower and Upper Integrals, and let the sum $f+g$ and the scalar multiple $cf$ of maps $f,g:[a,b]\to\mathbb{R}$ and $c\in\mathbb{R}$ be as in Pointwise Operations on Real-Valued Maps (i), with $I=[a,b]$. A map $h:[a,b]\to\mathbb{R}$ is bounded if there is $B\in\mathbb{R}$ with $|h(t)|\le B$ for every $t\in[a,b]$. Let $f,g:[a,b]\to\mathbb{R}$ be bounded maps that are integrable on $[a,b]$, and let $c\in\mathbb{R}$.

(i) $f+g$ is bounded and integrable on $[a,b]$, and $\int_{a}^{b}(f+g)=\int_{a}^{b}f+\int_{a}^{b}g$.

(ii) $cf$ is bounded and integrable on $[a,b]$, and $\int_{a}^{b}cf=c\int_{a}^{b}f$.

(iii) If $f(t)\le g(t)$ for every $t\in[a,b]$, then $\int_{a}^{b}f\le\int_{a}^{b}g$.

(iv) If $m,M\in\mathbb{R}$ and $m\le f(t)\le M$ for every $t\in[a,b]$, then $m\,(b-a)\le\int_{a}^{b}f\le M\,(b-a)$.

(v) If $B\in\mathbb{R}$ and $|f(t)|\le B$ for every $t\in[a,b]$, then $\bigl|\int_{a}^{b}f\bigr|\le B\,(b-a)$.

(vi) The constant map $[a,b]\to\mathbb{R}$ with value $c$ is bounded and integrable on $[a,b]$, and its integral is $\int_{a}^{b}c\,dt=c\,(b-a)$.