Goal: the following steps will guide you to prove proposition 1.2.2: Let a bounded and Riemann integrable function $f: [a, b] \rightarrow \mathbb{R}$, then \begin{equation*} \int_a^b f(x)dx = \overline{\int_a^b} f(x)dx. \end{equation*} 1.1 Using proposition 1.2.1, show that the following inequalities hold: $\forall P$, partition of $[a, b]$, \begin{align*} L(f, P) \le \int_a^b f(x)dx \le U(f, P), \\ -U(f, P) \le \int_a^b f(x)dx \le -L(f, P). \end{align*} 1.2 Prove the following inequality \begin{equation*} \left| \int_a^b f(x)dx - \overline{\int_a^b} f(x)dx \right| \le U(f, P) - L(f, P). \end{equation*}