I am reading about Riemann-Stieltjes Integrals in Carother's Real Analysis. I am trying to prove the following Theorem:
Theorem: $14.9$
Let $\alpha$ be continuous and increasing. Given $f$ $\in$ $R_{\alpha}([a,b])$ and $\epsilon >0 $.
There exists a step function $h$ on $[a,b]$ with $||h||_{\infty}$ $\leq$ $||f||_{\infty}$ such that $\int_{a}^{b}|f - h| d\alpha$ $<$ $\epsilon$.
Carother's uses the Riemann conditions to obtain a partition $P$ such that $U(f:P) - L(f:P)$ $<$ $\epsilon$. Using this partition, Carother's selects $t_i$ $\in$ $[x_{i-1}, x_i)$ and define the step function $h(x) = f(t_{i})$ for $i = 1,2,3,...n$. It follows clearly that $||h||_{\infty}$ $\leq$ $\|f||_{\infty}$ however, he makes the following statement that confuses me and concludes the proof using the statement:
"Since $\alpha$ is continuous, we have h $\in$ $R_{\alpha}([a,b])$"
I tried showing this myself by showing $U(h:P) - L(h:p) = \sum_{i=1}^{n}(M_{i} - m_{i})\Delta\alpha_{i} < \epsilon$ where $M_{i}$ and $m_{i}$ are the sup and inf of interval $i$ respectively. However, if $h(x)$ is a step function we have $M_{i} = m_{i}$ which implies $\sum_{i=1}^{n}(M_{i} - m_{i})\Delta\alpha_{i} = 0$. Therefore, I would not even consider the continuity of $\alpha$. Evidently, this is incorrect so if anyone could help explain it would be greatly appreciated.
The step function is $h = \sum_{j=1}^n f(t_j) \phi_j$ where
$$\phi_j(x) = \begin{cases}1,&x_{j=1} \leqslant x < x_j\\0, & \text{otherwise} \end{cases}$$
By linearity of integration it is enough to show that every function $\phi_j$ is integrable.
For any partition $P = (a=y_0,y_1, \ldots,y_{m-1}, y_m = b)$, since $\alpha $ is increasing , we have for some $p,q$,
$$\tag{*}0 \leqslant U(P,\phi_j,\alpha) - L(P,\phi_j, \alpha) \leqslant (\,\alpha(y_p)-\alpha(y_{p-1})\,) + (\,\alpha(y_q)-\alpha(y_{q-1})\,) $$
where the RHS bound is attained when $x_{j-1} \in (y_{p-1},y_p]$ and $x_j \in (y_{q-1},y_q)$.
Since $\alpha$ is continuous on the compact interval $[a,b]$ it is uniformly continuous and there exists $\delta > 0$ such that $|\alpha(u) - \alpha(v)| < \epsilon /2 $ when $|u-v| < \delta$.
Choosing the partition $P$ with norm $\|P\| < \delta$ we satisfy the integrability condition
$$U(P,\phi_j,\alpha) - L(P,\phi_j, \alpha) < \epsilon$$