Is the supremum of addition and multiplication of functions the addition and multiplication of supremum of functions?

Question

Let $f$, $g$ and $h$ be three real functions defined in $X \subset \mathbb{R}$.

I would like to know if is it true that $$ \sup_{x \in X} |f(x)+g(x)h(x)| \leq \sup_{x \in X} |f(x)| + \sup_{x \in X} |g(x)| \sup_{x \in X} |h(x)| .$$

Answer 1

Yes, it's true by the triangular inequality: for all $x\in X$, \begin{align} |f(x)+g(x)h(x)|&\leq |f(x)|+|g(x)h(x)|=|f(x)|+|g(x)||h(x)|\\ &\leq \sup_{x\in X}|f(x)|+\sup_{x\in X}(|g(x)||h(x)|)\\ &\leq \sup_{x\in X}|f(x)|+\sup_{x\in X}|g(x)|\sup_{x\in X}|h(x)|. \end{align}