Exercise 8.1.8

Consider an arbitrary $𝜖>0$. We have that with some gauge ${\delta }_1(x)$, we have $|R(f,P)-A_1|<𝜖∕2$ for all $P$ which are ${\delta }_1(x)$-fine. Similarly we have ${\delta }_2(x)$ where $|R(f,P)-A_2|<𝜖∕2$. Let $\delta (x)=\min <!--\; FUNCTION\; APPLICATION\; -->\{{\delta }_1(x),{\delta }_2(x)\}$ and note that $\delta$ is also a gauge, and that any partition $P$ which is $\delta (x)$-fine is also ${\delta }_1(x)$-fine and ${\delta }_2(x)$-fine. Find a partition $P$ which is $\delta (x)$-fine.

We then have $|R(f,P)-A_1|<𝜖∕2$ and $|A_2-R(f,P)|<𝜖∕2$, so $|A_2-A_1|<𝜖$, for arbitrary $𝜖>0$. Therefore $A_1=A_2$.