Riemann integrability/Elementary properties/Fact/Proof/Exercise

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

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

   $m(b-a)\le {\int }_a^{b}f(t)dt=M(b-a).$

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

   ${\int }_a^{b}f(t)dt\le {\int }_a^{b}g(t)dt.$

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 $c\in ℝ$ we have ${\int }_a^{b}cf(t)dt=c{\int }_a^{b}f(t)dt$.

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