A question about infimum and supremum.

Question

Let $f,g : \mathbb{R}\rightarrow [0,1]$ two functions. My question is whether or not we have that

$$|\inf_{x} f(x) - \inf_{x} g(x)|\leq \sup_{x} |f(x) - g(x)|$$.

If it is not true in general, what are the cases that it is true?.

Thanks in advance.

Answer 1

Yes the inequality always holds.

Let $F:=\inf_x f(x)$, $G:=\inf_x g(x)$. We can assume w.l.o.g. that $F<G$.

For any $0<\varepsilon<G-F$ there exists a $y$ such that $$f(y)<F+\varepsilon < G \leq g(y)$$ so $$\sup_x |f(x)-g(x)| \geq \underbrace{g(y)}_{\geq G} - f(y) \geq G-F - \varepsilon .$$ Letting $\varepsilon\to 0$ shows the claim.