91Ó°ÊÓ

Suppose \(f:[a, b] \rightarrow \mathbf{R}\) is a bounded function. For \(n \in \mathbf{Z}^{+}\), let \(P_{n}\) denote the partition that divides \([a, b]\) into \(2^{n}\) intervals of equal size. Prove that \(L(f,[a, b])=\lim _{n \rightarrow \infty} L\left(f, P_{n},[a, b]\right)\) and \(U(f,[a, b])=\lim _{n \rightarrow \infty} U\left(f, P_{n},[a, b]\right)\)

Suppose \(f:[a, b] \rightarrow \mathbf{R}\) is Riemann integrable. Prove that $$ \int_{a}^{b} f=\lim _{n \rightarrow \infty} \frac{b-a}{n} \sum_{j=1}^{n} f\left(a+\frac{j(b-a)}{n}\right) $$

Suppose \(f:[a, b] \rightarrow \mathbf{R}\) is a bounded function and \(c \in(a, b) .\) Prove that \(f\) is Riemann integrable on \([a, b]\) if and only if \(f\) is Riemann integrable on \([a, c]\) and \(f\) is Riemann integrable on \([c, b] .\) Furthermore, prove that if these conditions hold, then $$ \int_{a}^{b} f=\int_{a}^{c} f+\int_{c}^{b} f $$

Suppose \(f:[a, b] \rightarrow \mathbf{R}\) is Riemann integrable. Define \(F:[a, b] \rightarrow \mathbf{R}\) by $$ F(t)=\left\\{\begin{array}{ll} 0 & \text { if } t=a \\ \int_{a}^{t} f & \text { if } t \in(a, b] . \end{array}\right. $$

Suppose \(f:[a, b] \rightarrow \mathbf{R}\) is Riemann integrable. Prove that \(|f|\) is Riemann integrable and that $$ \left|\int_{a}^{b} f\right| \leq \int_{a}^{b}|f| $$

Suppose \(f:[a, b] \rightarrow \mathbf{R}\) is an increasing function, meaning that \(c, d \in[a, b]\) with \(c

Suppose \(f_{1}, f_{2}, \ldots\) is a sequence of Riemann integrable functions on \([a, b]\) such that \(f_{1}, f_{2}, \ldots\) converges uniformly on \([a, b]\) to a function \(f:[a, b] \rightarrow \mathbf{R}\). Prove that \(f\) is Riemann integrable and $$ \int_{a}^{b} f=\lim _{n \rightarrow \infty} \int_{a}^{b} f_{n} $$