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I appreciate if you could give me some hints on how to prove that :

$$\sup(|f|) - \inf(|f|) \le \sup(f) - \inf(f)$$

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Here is a strategy. Since you asked for hints, I let you check the details.

This is not really a fact about functions. It is rather a property of subsets $S$ of $\mathbb{R}$. So take such a set $S$ and denote $|S|:=\{|s|\;;\;s\in S\}$. We want $$ \sup |S|-\inf|S|\leq \sup S-\inf S. $$

We have an alternative.

Case 1: $\sup |S|=\sup S$. Clearly $\inf S\leq \inf |S|$. So the inequality holds.

Case 2: $\sup|S|=-\inf S$. Check that $-\inf|S|\leq \sup S$. The inequality follows.

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