# Proof

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:

### We will show that the upper Riemann integral is well-defined.

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:

$\int_{a}^{b~*}f(x)dx:=\inf D.$

### We will show that the lower Riemann integral is well-defined.

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:

$\int_{a~*}^{b}f(x)dx:=\sup D'.$

Github: ### References

#### Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983