A partition of \(\color{DarkOrchid}{L}\) is a finite subset \(\color{DarkOrchid}{P \subseteq L}\) such that \(\color{DarkOrchid}{P}\) contains the endpoints \(\color{DarkOrchid}{a}\) and \(\color{DarkOrchid}{b}\). We usually write a partition by ordering its points. For example, if \(\color{DarkOrchid}{P}\) has \(\color{DarkOrchid}{n}\) points, then we would write: \(\color{DarkOrchid}{\color{DarkOrchid}{P \equiv (a = x_0 < \ldots < x_n = b)}.}\) A partition is to be thought of as corresponding to a subdivision of \(\color{DarkOrchid}{L}\) into pieces joind by their endpoints: \(\color{DarkOrchid}{L = [x_0, x_1] \cup \ldots \cup [x_{n-1}, x_n]}\). The set of partitions of \(\color{DarkOrchid}{L}\) is denoted by \(\color{DarkOrchid}{\Delta_L}\).

Let \(\color{DarkOrchid}{P \equiv (a = x_0 < \ldots < x_n = b)}\) be a partition of \(\color{DarkOrchid}{[a,b]}\). We will show that \(\color{DarkOrchid}{L(f,P) \leq F(b) - F(a) \leq U(f, P)}\). By telescoping, we can write \(\color{DarkOrchid}{F(b) - F(a) = (F(x_n) - F(x_{n-1})) + \ldots + (F(x_1) - F(x_0)) = \sum_{j=1}^n F(x_j) - F(x_{j-1}). }\) By the Mean Value Theorem applied to the interval \(\color{DarkOrchid}{[x_{j-1} - x_j]}\), there exists some \(\color{DarkOrchid}{t_j \in (x_{j-1} - x_j)}\) such that \(\color{DarkOrchid}{F(x_j) - F(x_{j-1}) = F'(t_j)(x_j - x_{j-1}).}\) Now \(\color{DarkOrchid}{F'(t_i) = f(t_i)}\), so we obtain \(\color{DarkOrchid}{F(b) - F(a) = \sum_{j=1}^n (x_j - x_{j-1}) f(t_i).}\) However, since \(\color{DarkOrchid}{\inf_{[x_{i-1}, x_i}] f\ \leq\ f(t_i)\ \leq \sup_{[x_{i-1}, x_i]} f,}\) we obtain \(\color{DarkOrchid}{L(f,P) \leq F(b) - F(a) \leq U(f, P)}\).