Difference of the sup

Question

Let $f, g:A\to \mathbb R$ two functions. Is it true that

$$\sup_{x\in A}|f(x)|-\sup_{x\in A}|g(x)|\leq \sup_{x\in A}|f(x)-g(x)|?$$

Answer 1

Yes. Since by the triangle inequality we have $|f(x)|\leq |f(x)-g(x)|+|g(x)|$ for all $x\in A$ we conclude that $\sup_{x\in A}|f(x)|\leq \sup_{x\in A}(|f(x)-g(x)|+|g(x)|)$. For non negative functions supremum of sum is not bigger than sum of supremums, so $\sup_{x\in A}(|f(x)-g(x)|+|g(x)|)\leq \sup_{x\in A}|f(x)-g(x)|+\sup_{x\in A}|g(x)|$.