Let $E$ be a linear normed vector space, $f$ a functional $f : E \rightarrow \mathbb{R} $ and $ S = \{g \in E : \|g\| = 1\} \subset E$.
Fix $x \in E$. I need to prove that $$\liminf_{g \in S, \ \alpha \rightarrow +0 } \frac{f(x+\alpha g)-f(x)}{\alpha} \leq \inf_{g \in S} (\liminf_{\alpha \rightarrow +0}\frac{f(x+\alpha g)-f(x)}{\alpha}) $$ I found this fact in my lecture notes and I have no idea of what's going on.