Stieltjes Integrable and series

I have the following conjecture that I would like to see proven or disproven:

Let $\{\alpha_i\}$ be a sequence of monotonically increasing functions on $[a,b]$. Suppose $f\in \mathscr{R}(\alpha_i), \forall i\in \mathbb{N}$ and $\sum \alpha_i$ is convergent. Now let $A=\sum_{i=1}^{\infty} \alpha_i$. Then is it true that $f\in \mathscr{R}(A)$?

If it is not true, then what if $f$ is continuous at every discontinuity of each $\alpha_i$?