Riemann-Stieltjes Integrability and Convergent Series Let $\alpha_{n=1}^{\infty}$ be a sequence of monotonically increasing functions on $[a.b]$ such that the series $\sum_{n=1}^{\infty}\alpha_{n}(a)$ and $\sum_{n=1}^{\infty}\alpha_{n}(b)$ converge.
Suppose that $f \in \mathcal{R}(\alpha_{n})$ $\forall n \in \mathbb{N}$. Show that $f \in \mathcal{R}(\alpha)$ and that:
$\displaystyle\int_{0}^{1}f\mathrm{d\alpha} = \sum_{n=1}^{\infty}\int_{0}^{1}f\mathrm{d\alpha_{n}}$

I'm having mostly trouble with showing that $f \in \mathcal{R}(\alpha)$ (if this can be shown, then the equality with the integrals follows easily). We have that if $f \in \mathcal{R}(\alpha_{1})$ and $f \in \mathcal{R}(\alpha_{2})$, then it follows that $f \in \mathcal{R}(\alpha_{1}+\alpha_{2})$. However, I don't know how to generalize this notion when we have an convergent series of infinite terms $\alpha_{n}$.
What would be the way to approach this problem? Thank you for the help.

To show $f\in R(\alpha):$ Let $\epsilon > 0.$ Choose $N$ such that
$$(1)\,\,\,\,\,\sum_{n>N}(\alpha_n(b)-\alpha_n(a)) < \epsilon.$$ Now choose a partition $\{x_0, \dots ,x_l\}$ such that
$$(2)\,\,\,\,\sum_{k=1}^{l}(M_k-m_k) (\alpha_n(x_k)-\alpha_n(x_{k-1})) < \epsilon/N, n=1,\dots,N.$$
We can do this by first finding a partition that works for $\alpha_1,$ then refining it to work for $\alpha_2,$ etc. Since there are only finitely many refinements, we're OK. We want to show this partition works for $\alpha = \sum_n \alpha_n.$ Let' see:
$$\sum_{k=1}^{l}(M_k-m_k)\sum_{n=1}^{\infty}(\alpha_n(x_k)-\alpha_n(x_{k-1})) = \sum_{n=1}^{\infty}\sum_{k=1}^{l}(M_k-m_k)(\alpha_n(x_k)-\alpha_n(x_{k-1})).$$
Switching the order of summation is fine because everything is nonnegative. In the last double sum, split the outer sum into $\sum_{n=1}^{N} $ and $\sum_{n>N}.$ The first one is $<\epsilon $ by (2). The second one is no more that $(2\sup|f|)\epsilon$ by (1).
