Riemann integrability/Elementary properties/Fact

Let ${}I=[a,b]$ denote a compact interval, and let ${}f,g\colon I\rightarrow \mathbb {R}$ denote Riemann-integrable functions. Then the following statements hold.

1. If ${}m\leq f(x)\leq M$ holds for all ${}x\in I$, then ${}m(b-a)\leq \int _{a}^{b}f(t)\,dt\leq M(b-a)$ holds.
2. If ${}f(x)\leq g(x)$ holds for all ${}x\in I$, then ${}\int _{a}^{b}f(t)\,dt\leq \int _{a}^{b}g(t)\,dt$ holds.
3. The sum ${}f+g$ is Riemann-integrable, and the identity ${}\int _{a}^{b}(f+g)(t)\,dt=\int _{a}^{b}f(t)\,dt+\int _{a}^{b}g(t)\,dt$ holds.
4. For ${}c\in \mathbb {R}$ we have ${}\int _{a}^{b}(cf)(t)\,dt=c\int _{a}^{b}f(t)\,dt$.
5. The functions ${}{\max {\left(f,g\right)}}$ and ${}{\min {\left(f,g\right)}}$ are Riemann-integrable.
6. The function ${}\vert {f}\vert$ is Riemann-integrable.
7. The product ${}fg$ is Riemann-integrable.

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