Theorem: Product of Functions of Bounded Variation is of Bounded Variation

Let $a, b$ be real numbers with $a < b$.
Let $f, g : \closedint a b \to \R$ be functions of bounded variation.
Let $\map {V_f} {\closedint a b}$ and $\map {V_g} {\closedint a b}$ be the total variations of $f$ and $g$ respectively. 
Then the pointwise product $f \cdot g$ is of bounded variation with:
:$\map {V_{f \cdot g} } {\closedint a b} \le A \map {V_f} {\closedint a b} + B \map {V_g} {\closedint a b}$
where:
:$\map {V_{f \cdot g} } {\closedint a b}$ denotes the total variation of $f \cdot g$ on $\closedint a b$ 
:$A, B$ are non-negative real numbers.

Proof:
For each finite subdivision $P$ of $\closedint a b$, write: 
:$P = \set {x_0, x_1, \ldots, x_n }$
with:
:$a = x_0 < x_1 < x_2 < \cdots < x_{n - 1} < x_n = b$
By Function of Bounded Variation is Bounded: 
:$f$ and $g$ are bounded. 
So, there exists $A, B \in \R$ such that: 
:$\size {\map f x} \le B$
:$\size {\map g x} \le A$
for all $x \in \closedint a b$.
Then:
{{begin-eqn}}
{{eqn	| l = \map {V_{f \cdot g} } {P ; \closedint a b}
	| r = \sum_{i \mathop = 1}^n \size {\map {\paren {f \cdot g} } {x_i} - \map {\paren {f \cdot g} } {x_{i - 1} } }
	| c = using the notation from the definition of bounded variation
}}
{{eqn	| r = \sum_{i \mathop = 1}^n \size {\map f {x_i} \map g {x_i} - \map f {x_{i - 1} } \map g {x_{i - 1} } }
}}
{{eqn	| r = \sum_{i \mathop = 1}^n \size {\map f {x_i} \map g {x_i} - \map f {x_{i - 1} } \map g {x_i} + \map f {x_{i - 1} } \map g {x_i} - \map f {x_{i - 1} } \map g {x_{i - 1} } }
}}
{{eqn	| o = \le
	| r = \sum_{i \mathop = 1}^n \size {\map f {x_i} \map g {x_i} - \map f {x_{i - 1} } \map g {x_i} } + \sum_{i \mathop = 1}^n \size {\map f {x_{i - 1} } \map g {x_i} - \map f {x_{i - 1} } \map g {x_{i - 1} } }
	| c = Triangle Inequality
}}
{{eqn	| r = \sum_{i \mathop = 1}^n \size {\map g {x_i} } \size {\map f {x_i} - \map f {x_{i - 1} } } + \sum_{i \mathop = 1}^n \size {\map f {x_{i - 1} } } \size {\map g {x_i} - \map g {x_{i - 1} } }
}}
{{eqn	| o = \le
	| r = A \sum_{i \mathop = 1}^n \size {\map f {x_i} - \map f {x_{i - 1} } } + B \sum_{i \mathop = 1}^n \size {\map g {x_i} - \map g {x_{i - 1} } }
	| c = since $\size {\map g {x_i} } \le A$ and $\size {\map f {x_{i - 1} } } \le B$
}}
{{eqn	| r = A \map {V_f} {P ; \closedint a b} + B \map {V_g} {P ; \closedint a b}
}}
{{end-eqn}}
Since $f$ and $g$ are of bounded variation, there exists $M, K \in \R$ such that: 
:$\map {V_f} {P ; \closedint a b} \le M$
:$\map {V_g} {P ; \closedint a b} \le K$
for all finite subdivisions $P$.
We therefore have: 
:$\map {V_{f \cdot g} } {P ; \closedint a b} \le A M + B K$
so $f \cdot g$ is of bounded variation.
Further, we have: 
{{begin-eqn}}
{{eqn	| l = \map {V_{f \cdot g} } {\closedint a b}
	| r = \sup_P \paren {\map {V_{f \cdot g} } {P ; \closedint a b} }
	| c = {{Defof|Total Variation of Real Function on Closed Bounded Interval}}
}}
{{eqn	| o = \le
	| r = \sup_P \paren {A \map {V_f} {P ; \closedint a b} } + \sup_P \paren {B \map {V_g} {P ; \closedint a b} }
}}
{{eqn	| r = A \sup_P \paren {\map {V_f} {P ; \closedint a b} } + B \sup_P \paren {\map {V_g} {P ; \closedint a b} }
	| c = Multiple of Supremum
}}
{{eqn	| r = A \map {V_f} {\closedint a b} + B \map {V_g} {\closedint a b}
		| c = {{Defof|Total Variation of Real Function on Closed Bounded Interval}}
}}
{{end-eqn}}
{{qed}}