$f, g : [a, b] \to R$ be of bounded variation. Then $fg$ is of bounded variation

Question

Claim

$f, g : [a, b] → R$ be of bounded variation. Then show that $fg$ is of bounded variation.

---

To prove above claim, I would like to derive the fact such as $\mid f(x_i) g(x_i)-f(x_{i-1})g(x_{i-1})\mid\le\mid f(x_i) -f(x_{i-1})\mid \mid g(x_i)-g(x_{i-1})\mid$ (*)

any advice to handle this absolute inequality so that I could prove (*)?

Answer 1

Try this: $∣f(x_i)g(x_i)−f(x_{i−1})g(x_{i−1})∣ = |f(x_i)g(x_i)−f(x_{i−1})g(x_{i−1}) +f(x_i)g(x_{i-1})-f(x_i)g(x_{i-1})| = |f(x_i)(g(x_i)-g(x_{i-1})) + (f(x_i) −f(x_{i−1}))g(x_{i−1}) |$