Uniform Partitions and Order Bounds for the Riemann Integral

Let $p,q$ be real numbers with $p<q$ in the order of the ordered field $\mathbb{R}$, let $[p,q]$ be the closed interval determined by $p$ and $q$, and let $\mathbb{N}$ be the set of natural numbers, each $n\in\mathbb{N}$ being identified with its image in $\mathbb{R}$ under the canonical map.

1. (Uniform partitions) For every $n\in\mathbb{N}$ the points $x_0=p$ and

$$x_i=p+i\,\frac{q-p}{n}\qquad(i=1,\dots,n)$$

form a partition $P_n=(x_0,x_1,\dots,x_n)$ of $[p,q]$ with mesh $|P_n|=(q-p)/n$. Moreover, for every real $\delta>0$ there exists $n\in\mathbb{N}$ with $(q-p)/n<\delta$.

2. (Order bounds) Let $f:[p,q]\to\mathbb{R}$ be Riemann integrable on $[p,q]$ and let $m,M\in\mathbb{R}$ satisfy $m\le f(u)\le M$ for every $u\in[p,q]$. Then

$$m\,(q-p)\le\int_p^q f(u)\,du\le M\,(q-p) .$$