Riemann integrability/Elementary properties/Fact/Proof/Exercise

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

1. If ${}m\leq f(t)\leq M$ for all ${}t\in I$, then

   ${}m(b-a)\leq \int _{a}^{b}f(t)dt=M(b-a)\,.$

2. If ${}f(t)\leq g(t)$ for all ${}t\in I$, then

   ${}\int _{a}^{b}f(t)dt\leq \int _{a}^{b}g(t)dt\,.$

3. We have

   ${}\int _{a}^{b}f(t)+g(t)dt=\int _{a}^{b}f(t)dt+\int _{a}^{b}g(t)dt\,.$

4. For ${}c\in \mathbb {R}$ we have ${}\int _{a}^{b}cf(t)dt=c\int _{a}^{b}f(t)dt$.

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