I have a question regarding the proof of Theorem 9.8 from Mathematical Analysis by Tom Apostol below:
Theorem 9.8: Let $\alpha$ be of a bounded variation on $[a,b]$. Assume that each term of the sequence ${f_n}$ is a real-valued functtion such that $f_n \in R(\alpha)$ on $[a,b]$ for each $n=1,2,...$ Assume that $f_n \to f$ uniformly on $[a,b]$ and define $g_n(x) = \int_a^x \! f_n(t) \, \mathrm{d}\alpha (t)$ if $x\in [a,b], n=1,2,...$ Then we have:
- $f\in R(\alpha)$ on $[a,b]$
- $g_n \to g$ uniformly on $[a,b]$, where $g(x) = \int_a^x \! f(t) \, \mathrm{d}\alpha (t)$
Proof: We can assume that $\alpha$ is increasing with $\alpha(a)<\alpha (b)$. To prove (a), we will show what $f$ satisfies Riemann's condition with respect to $\alpha$ on $[a,b]$ ... (I omit the rest of the proof, there is no mention of generalizing $\alpha$ to the general case.
We know that $\alpha$ is monotonic $\implies \alpha$ is of bounded variation. How come we can assume $\alpha$ is increasing?