Theorem: Sum of Riemann-Stieltjes Integrals on Adjacent Intervals/Part to Whole

Let $a < c < b$ be real numbers.
Let $f, \alpha$ be real functions that are bounded on $\closedint a b$
Suppose that $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $\closedint a c$, and also on $\closedint c b$.
Then, $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $\closedint a b$, and:
:$\ds \int_a^b f \rd \alpha = \int_a^c f \rd \alpha + \int_c^b f \rd \alpha$

Proof:
Let $\epsilon > 0$ be arbitrary.
By definition of the Riemann-Stieltjes integral, let $P'_\epsilon$ be a subdivision of $\closedint a c$ and $P' '_\epsilon$ be a subdivision of $\closedint c b$ such that:
:For every $P'$ finer than $P'_\epsilon$, $\ds \size {\map S {P', f, \alpha} - \int_a^c f \rd \alpha} < \frac \epsilon 2$
:For every $P' '$ finer than $P' '_\epsilon$, $\ds \size {\map S {P' ', f, \alpha} - \int_c^b f \rd \alpha} < \frac \epsilon 2$
Define $P_\epsilon := P'_\epsilon \cup P' '_\epsilon$.
Since:
* $P_\epsilon \subseteq \closedint a c \cup \closedint c b = \closedint a b$
* $a \in P'_\epsilon \subseteq P_\epsilon$
* $b \in P' '_\epsilon \subseteq P_\epsilon$
it follows that $P_\epsilon$ is a subdivision of $\closedint a b$.
Let $P = \set {x_0, x_1, \dotsc, x_{n - 1}, x_n}$ be a subdivision of $\closedint a b$ that is finer than $P_\epsilon$.
Since $c \in P'_\epsilon \subseteq P_\epsilon \subseteq P$, there must be some $m \in {1, 2, \dotsc, n - 2, n - 1}$ such that:
:$x_m = c$
For each $\ell \in \set {0, 1, \dotsc, n - m - 1, n - m}$, define:
:$y_\ell = x_{\ell + m}$
Then, define:
:$P' := P \cap \closedint a c = \set {x_0, x_1, \dotsc, x_{m - 1}, x_m}$
:$P' ' := P \cap \closedint c b = \set {y_0, y_1, \dotsc, y_{n - m - 1}, y_{n - m}}$
It follows that:
:$P'$ is a subdivision of $\closedint a c$ that is finer than $P'_\epsilon$
:$P' '$ is a subdivision of $\closedint c b$ that is finer than $P' '_\epsilon$
We have:
{{begin-eqn}}
{{eqn | l = \map S {P, f, \alpha}
      | r = \sum_{k \mathop = 1}^n \map f {t_k} \paren {\map \alpha {x_k} - \map \alpha {x_{k - 1} } }
      | c = {{Defof|Riemann-Stieltjes Sum}}
}}
{{eqn | r = \sum_{k \mathop = 1}^m \map f {t_k} \paren {\map \alpha {x_k} - \map \alpha {x_{k - 1} } } + \sum_{k \mathop = m + 1}^n \map f {t_k} \paren {\map \alpha {x_k} - \map \alpha {x_{k - 1} } }
      | c = Indexed Summation over Adjacent Intervals
}}
{{eqn | r = \sum_{k \mathop = 1}^m \map f {t_k} \paren {\map \alpha {x_k} - \map \alpha {x_{k - 1} } } + \sum_{\ell \mathop = 1}^{n - m} \map f {t_{\ell + m} } \paren {\map \alpha {x_{\ell + m} } - \map \alpha {x_{\ell + m - 1} } }
      | c = Indexed Summation over Translated Interval
}}
{{eqn | r = \sum_{k \mathop = 1}^m \map f {t_k} \paren {\map \alpha {x_k} - \map \alpha {x_{k - 1} } } + \sum_{\ell \mathop = 1}^{n - m} \map f {t_{\ell + m} } \paren {\map \alpha {y_\ell} - \map \alpha {y_{\ell - 1} } }
      | c = Definition of $y_\ell$
}}
{{eqn | r = \map S {P', f, \alpha} + \map S {P' ', f, \alpha}
      | c = {{Defof|Riemann-Stieltjes Sum}}, as $y_{\ell - 1} = x_{\ell + m - 1} \le t_{\ell + m} \le x_{\ell + m} = y_\ell$
}}
{{end-eqn}}
Therefore:
{{begin-eqn}}
{{eqn | l = \size {\map S {P, f, \alpha} - \paren {\int_a^c f \rd \alpha + \int_c^b f \rd \alpha} }
      | r = \size {\map S {P', f, \alpha} - \int_a^c f \rd \alpha + \map S {P' ', f, \alpha} - \int_c^b f \rd \alpha}
      | c = Above
}}
{{eqn | o = \le
      | r = \size {\map S {P', f, \alpha} - \int_a^c f \rd \alpha} + \size {\map S {P' ', f, \alpha} - \int_c^b f \rd \alpha}
      | c = Triangle Inequality for Real Numbers
}}
{{eqn | o = <
      | r = \frac \epsilon 2 + \frac \epsilon 2
      | c = Definitions of $P'_\epsilon$ and $P' '_\epsilon$
}}
{{eqn | r = \epsilon
}}
{{end-eqn}}
As $P \supseteq P_\epsilon$ and $\epsilon > 0$ were arbitrary, by the definition of the Riemann-Stieltjes integral:
:$\ds \int_a^b f \rd \alpha = \int_a^c f \rd \alpha + \int_c^b f \rd \alpha$
{{qed}}