(related to Proposition: Riemann Upper and Riemann Lower Integrals for Bounded Real Functions)
Let \([a,b]\) be a closed real interval and let \(f\) be bounded on \([a,b]\) and let \(\psi\) be a step function defined on that interval.
The proof goes in two steps:
Obviously, if by hypothesis
\[\phi(x)\ge f(x)\text{ for all }x\in[a,b]\]
then \(\phi\) is bounded from below by \(f\). The set \(D\) of all Riemann integrals of such functions:
\[D:=\left\{\int_{a}^b\phi(x)dx:~~\phi(x)\ge f(x)\text{ for all }x\in[a,b]\text{ and }\phi\text{ is a step function over }[a,b]\right\}\]
is therefore a subset of real numbers, which is bounded from below. From the infimum property of real numbers, it follows that the following infimum exists:
\[\inf D:=\inf\left\{\int_{a}^b\phi(x)dx:~~\phi(x)\ge f(x)\text{ for all }x\in[a,b]\text{ and }\phi\text{ is a step function over }[a,b]\right\}.\]
The definition of infimum shows that it is unique. Therefore, we the upper Riemann integral of the bounded function \(f\) is well-defined:
\[\int_{a}^{b~*}f(x)dx:=\inf D.\]
Similarly, if by hypothesis
\[\phi(x)\le f(x)\text{ for all }x\in[a,b]\]
then \(\phi\) is bounded from above by \(f\). The set \(D\) of all Riemann integrals of such functions:
\[D':=\left\{\int_{a}^b\phi(x)dx:~~\phi(x)\le f(x)\text{ for all }x\in[a,b]\text{ and }\phi\text{ is a step function over }[a,b]\right\}\]
is therefore a subset of real numbers, which is bounded from above. From the supremum property of real numbers, it follows that the following supremum exists:
\[\inf D':=\inf\left\{\int_{a}^b\phi(x)dx:~~\phi(x)\le f(x)\text{ for all }x\in[a,b]\text{ and }\phi\text{ is a step function over }[a,b]\right\}.\]
The definition of supremum shows that it is unique. Therefore, we the lower Riemann integral of the bounded function \(f\) is well-defined:
\[\int_{a~*}^{b}f(x)dx:=\sup D'.\]