(Basic properties of the Riemann integral). Let $[a, b]$ be an interval, and let $f, g:[a, b] \rightarrow \mathbf{R}$ be Riemann integrable. Establish the following statements:
(1) (Linearity) For any real number $c, c f$ and $f+g$ are Riemann integrable, with $\int_a^b c f(x) d x=c \cdot \int_a^b f(x) d x$ and $\int_a^b f(x)+ g(x) d x=\int_a^b f(x) d x+\int_a^b g(x) d x$.
(2) (Monotonicity) If $f \leq g$ pointwise (i.e. $f(x) \leq g(x)$ for all $x \in[a, b])$ then $\int_a^b f(x) d x \leq \int_a^b g(x) d x$.
(3) (Indicator) If $E$ is a Jordan measurable of $[a, b]$, then the indicator function $1_E:[a, b] \rightarrow \mathbf{R}$ (defined by setting $1_E(x):=$ 1 when $x \in E$ and $1_E(x):=0$ otherwise) is Riemann integrable, and $\int_a^b 1_E(x) d x=m(E)$.