# Proof

(related to Proposition: Riemann Integral for Step Functions)

Let $$\phi\in T[a,b]$$ be a step function with respect to the partition $$a=x_0 < x_1 < \ldots < x_{n-1} < x_n=b$$. We want to show that the definition of the Riemann integral $\int_a^b\phi(x)dx:=\sum_{i=1}^nc_i(x_i-x_{i-1})$ does not depend on the specific choice of the partition $$a=x_0 < x_1 < \ldots < x_{n-1} < x_n=b$$, i.e. for any such partition, the above sum takes the same value.

### Case 1

Assume

$\begin{array}{rcl} S&:=&a=x_0 < x_1 < \ldots < x_{n-1} < x_n=b\\ T&:=&a=t_0 < t_1 < \ldots < t_{m-1} < x_m=b\\ \end{array}$

are two different partitions of the closed interval $$[a,b]$$ such that

$\begin{array}{rcl} \phi (x)=c_i\quad&\quad&\text{for all } x\in(x_{i-1},x_i),\quad i=1,\ldots,n,\\ \phi (x)=d_j\quad&\quad&\text{for all } x\in(t_{j-1},t_j),\quad j=1,\ldots,m.\\ \end{array}$

and assume that $$S\subset T$$. This means that for each $$i$$ there is an index $$j_i$$ such that $$x_i=t_{j_i}$$. Then we have

$x_{i-1}=t_{j_{i-1}} < t_{j_{i-1}+ 1} < t_{j_{i-1}+ 2} < \ldots < t_{j_i}=x_i\quad\quad\text{for }1\le i\le n.$ and $c_i=d_j\quad\quad\text{for }j_{i-1} < i \le j_i,\quad 1\le i\le n.$ It follows $\sum_{i=1}^{n}c_i(x_i-x_{i-1})=\sum_{i=1}^n\sum_{k=j_{i-1}+1}^{j_i}d_k(t_k-t_{k-1}).$

In other words, the Riemann integrals with respect to the partitions $$S$$ and $$T$$ equal each other.

### Case 2

Now assume arbitrary partitions $$S$$ and $$T$$ and that $$U=S\cup T$$. Then we have $$S\subset U$$ and $$T\subset U$$. According to case 1, the Riemann integrals with respect to the partitions $$S$$ and $$U$$ equal each other, as well as the Riemann integrals with respect to the partitions $$T$$ and $$U$$ equal each other. Thus, the Riemann integrals with respect to the partitions $$S$$ and $$T$$ also equal each other. In other words, the Riemann integral of the step function does not depend on the specific choice of the partition.

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### References

#### Bibliography

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