i've posted this question here but i probably realized that the problem is my lack of skills in manipulating sup and inf definition properly. I tried to formalize my problem:
Let $S$ a set and let $f$ a function defined from $S$ to $\mathbb{R}$ let also suppose that $\alpha \in \mathbb{R}$ is a value such that $\forall s \in S$ we have $\alpha \leq f(s)$, also let's say that for each $\epsilon > 0$ there's an $s \in S$ such that $f(s) < \alpha + \epsilon$ can we deduce from this that
$$\alpha = \inf_{s \in S} f(s)$$ ?
My attempt is based on the definition of $sup/inf$. Of course $\alpha$ is a lower bound for $f(s)$ by hypothesis, on the other side let's take arbitrary $\beta > \alpha$ and let's show that $\beta$ cannot be a lower bound. If we define
$$\epsilon = \beta - \alpha > 0 \Rightarrow \beta = \alpha + \epsilon$$, and by assumption there's $s \in S$ such that $f(s) < \alpha + \epsilon = \beta$ so $$\beta > f(s)$$ and then this cannot be a lower bound, so $\alpha$ is the $inf$ for $f(s)$. Is this reasoning correct and formal?