Theorem: Linear Combination of Riemann-Stieltjes Integrals/Integrator

Let $f, \alpha, \beta$ be real functions that are bounded on $\closedint a b$.
Let $c_1, c_2 \in \R$.
Suppose that $f$ is Riemann-Stieltjes integrable with respect both $\alpha$ and $\beta$ on $\closedint a b$ and:
:$\ds \int_a^b f \rd \alpha = A$
:$\ds \int_a^b f \rd \beta = B$
Then, $f$ is Riemann-Stieltjes integrable with respect to $\gamma : \closedint a b \to \R$ defined as:
:$\map \gamma x = c_1 \map \alpha x + c_2 \map \beta x$
and furthermore:
:$\ds \int_a^b f \rd \gamma = c_1 A + c_2 B$

Proof:
Let $\epsilon > 0$ be arbitrary.
By definition of the Riemann-Stieltjes integral, let $P'_\epsilon, P' '_\epsilon$ be subdivisions of $\closedint a b$ such that:
:For every $P$ finer than $P'_\epsilon$, $\size {\map S {P, f, \alpha} - A} < \dfrac \epsilon {\size {c_1} + \size {c_2} + 1}$
:For every $P$ finer than $P' '_\epsilon$, $\size {\map S {P, f, \beta} - B} < \dfrac \epsilon {\size {c_1} + \size {c_2} + 1}$
Define $P_\epsilon = P'_\epsilon \cup P' '_\epsilon$ to be a subdivision that is finer than both $P'_\epsilon$ and $P' '_\epsilon$.
Let $P = {x_0, \dotsc, x_n} \supseteq P_\epsilon$ be an arbitrary subdivision that is finer than $P_\epsilon$.
Clearly, $P$ is also finer than $P'_\epsilon$ and $P' '_\epsilon$.
Finally, fix the $t_1, \dotsc, t_k$ for the following Riemann-Stieltjes sum.
We have:
{{begin-eqn}}
{{eqn | l = \map S {P, f, \gamma}
      | r = \sum_{k \mathop = 1}^n \map f {t_k} \paren {\map \gamma {x_k} - \map \gamma {x_{k - 1} } }
      | c = {{Defof|Riemann-Stieltjes Sum}}
}}
{{eqn | r = \sum_{k \mathop = 1}^n \map f {t_k} \paren {\paren {c_1 \map \alpha {x_k} + c_2 \map \beta {x_k} } - \paren {c_1 \map \alpha {x_{k - 1} } + c_2 \map \beta {x_{k - 1} } } }
      | c = Definition of $\gamma$
}}
{{eqn | r = c_1 \sum_{k \mathop = 1}^n \map f {t_k} \paren {\map \alpha {x_k} - \map \alpha {x_{k - 1} } } + c_2 \sum_{k \mathop = 1}^n \map f {t_k} \paren {\map \beta {x_k} - \map \beta {x_{k - 1} } }
      | c = Linear Combination of Indexed Summations
}}
{{eqn | r = c_1 \map S {P, f, \alpha} + c_2 \map S {P, f, \beta}
      | c = {{Defof|Riemann-Stieltjes Sum}}
}}
{{end-eqn}}
And so:
{{begin-eqn}}
{{eqn | l = \size {\map S {P, f, \gamma} - \paren {c_1 A + c_2 B} }
      | r = \size {\paren {c_1 \map S {P, f, \alpha} + c_2 \map S {P, f, \beta} } - \paren {c_1 A + c_2 B} }
}}
{{eqn | r = \size {c_1 \paren {\map S {P, f, \alpha} - A} + c_2 \paren {\map S {P, f, \beta} - B} }
}}
{{eqn | o = \le
      | r = \size {c_1} \size {\map S {P, f, \alpha} - A} + \size {c_2} \size {\map S {P, f, \beta} - B}
      | c = Triangle Inequality for Real Numbers and Absolute Value Function is Completely Multiplicative
}}
{{eqn | o = \le
      | r = \size {c_1} \frac \epsilon {\size {c_1} + \size {c_2} + 1} + \size {c_2} \frac \epsilon {\size {c_1} + \size {c_2} + 1}
      | c = Definitions of $P'_\epsilon$ and $P' '_\epsilon$
}}
{{eqn | r = \frac {\size {c_1} + \size {c_2} } {\size {c_1} + \size {c_2} + 1} \epsilon
}}
{{eqn | o = <
      | r = \epsilon
}}
{{end-eqn}}
As $\epsilon > 0$ was arbitrary, by the definition of the Riemann-Stieltjes integral:
:$\ds \int_a^b f \rd \gamma = c_1 A + c_2 B$
{{qed}}