I'm trying to figure out the following detail from the proof of a proposition. But It's been a while since I've worked with inferior limits:
Let $E$ be a topological space, $a \in E$, $\lambda \in \mathbb{R}$, and $f:E \rightarrow \tilde{\mathbb{R}}$ a function. If there exists a neighborhood $V$ of $a$ such that
$$\lambda \leq \tilde{f(V)},$$
then
$$\lambda \leq \varliminf_{x \to a} f(x).$$
Could you give me an idea?