Comments on the Riemann Integral

Show that, in the above Dirichlet function example, $\left\{ f_{n}\right\}$ fails to converge to $f$ uniformly on $[0,1]$.

A partition $P^{\prime }$ of $[a, b]$ is called a refinement of a partition $P$ provided that each partition point of $P$ is also a partition point of $P^{\prime }$. For a bounded function $f:[a, b] \rightarrow \mathbb {R}$, show that under refinement lower Darboux sums do not decrease and upper Darboux sums do not increase.

Show that for a bounded function $f:[a, b] \rightarrow \mathbb {R}$, the left-hand side of (1) is no greater than the right-hand side.

Assume that $f:[a, b] \rightarrow \mathbb {R}$ is bounded. Show that $f:[a, b] \rightarrow \mathbb {R}$ is Riemann integrable if and only if there is a sequence $\left\{ P_{n}\right\}$ of partitions of $[a, b]$, each $P_{n+1}$ being a refinement of $P_{n}$, for which $\lim_{n \rightarrow \infty }\left[U\left(f, P_{n}\right)-L\left(f, P_{n}\right)\right] = 0$.

Use the preceding problem to show that a continuous function $f:[a, b] \rightarrow \mathbb {R}$, being uniformly continuous, is Riemann integrable.

Let $f:[0,1] \rightarrow \mathbb {R}$ be increasing. For a natural number $n$, define $P_{n}$ to be the partition of $[0,1]$ into $n$ subintervals of length $1 / n$. Show that $U\left(f, P_{n}\right)-L\left(f, P_{n}\right) \leq 1 / n[f(1)-f(0)]$. Use Problem 4 to show that $f:[0,1] \rightarrow \mathbb {R}$ is Riemann integrable.

Let $\left\{ f_{n}:[a, b] \rightarrow \mathbb {R}\right\}$ be a uniformly bounded sequence of Riemann integrable functions that converges uniformly to $f:[a, b] \rightarrow \mathbb {R}$. Show that $f:[a, b] \rightarrow \mathbb {R}$ also is Riemann integrable, and