Difference of Functions of Bounded Variation is of Bounded Variation

Theorem
Let $a, b$ be real numbers with $a < b$.

Let $f, g : \closedint a b \to \R$ be functions of bounded variation.

Let $V_f$ and $V_g$ be the total variations of $f$ and $g$ respectively.

Then $f - g$ is of bounded variation with:


 * $V_{f - g} \le V_f + V_g$

where $V_{f - g}$ denotes the total variation of $f - g$.

Proof
By Multiple of Function of Bounded Variation is of Bounded Variation, we have that:


 * $-g$ is of bounded variation.

So, by Sum of Functions of Bounded Variation is of Bounded Variation, we have that:


 * $f + \paren{-g} = f - g$ is of bounded variation

with:


 * $V_{f - g} \le V_f + V_{-g}$

where $V_{-g}$ is the total variation of $-g$.

We have, by Multiple of Function of Bounded Variation is of Bounded Variation:

so:


 * $V_{f - g} \le V_f + V_g$