The question was given to me as follows: Let $\alpha_1$ and $\alpha_2$ be increasing functions on $[a,b]$.
(a) Prove that $\alpha_1 + \alpha_2$ is increasing. (I did this using inequalities)
(b) Let $f \in \Re_{\alpha_1}[a,b] \cap \Re_{\alpha_2}[a,b].$ Prove that $f \in \Re_{\alpha_1 + \alpha_2}[a,b]$ and $$\int_a^b f d(\alpha_1 + \alpha_2) = \int_a^b f d\alpha_1 + \int_a^b f d\alpha_2$$
This is where I'm lost now. I know since $\alpha_1 + \alpha_2$ is increasing that I can use some things I know but I'm not sure which. I am thinking I can maybe use the Riemann condition? Since I know that $f \in \Re_{\alpha_1}[a,b] \cap \Re_{\alpha_2}[a,b]$ that means that for every $\varepsilon > 0$, there exists a partition $P$ of $[a,b]$ such that $U(f,P) - L(f,P) < \varepsilon$. Can I now work with this? Or is another concept more appropriate?
Using Riemann sums: $S(f, P, T, \alpha_1 + \alpha_2) = S(f, P, T, \alpha_1) + S(f, P, T, \alpha_2)$ for any partition $P$ and any choice of valid tags $T$ for $P.$ QED
As for Darboux's sums, prove that $U(f, P, \alpha_1 + \alpha_2) \leq U(f, P, \alpha_1) + U(f, P, \alpha_2)$ and that $U(f, P, \alpha_1) + L(f, P, \alpha_2) \leq L(f, P, \alpha_1 + \alpha_2).$ And then use an $\dfrac{\varepsilon}{2}$-argument