Clarification on proof about Lipschitz function composition

Question

The solution is found here: sum and product of Lipschitz functions

I'm not sure from where the following line $$|f(x)g(x) - f(x)g(y)| \leq M|g(x) - g(y)|$$

comes from.

I tried looking at assumption that $f$ was Lipschitz, but I'm not sure where this conclusion came from.

Answer 1

Since $|f(x)|\leq M$ we have, $$|f(x)g(x) - f(x)g(y)| = |f(x)||g(x) - g(y)|\leq M|g(x)-g(y)|.$$