Let $X,Y$ be two Hausdorff spaces and $F:X\to 2^Y$ be a multi-valued mapping. We says that $F$ is *lower semicontinuous* at $x_0\in X$ if for each $y_0\in F(x_0)$ and any neighborhood $U\in \mathcal N(y_0)$, there exist a neighborhood $V$ of $x_0$ such that
$$
F(x) \cap U\neq \emptyset \quad\text{for all}\quad x \in V.
$$

In Aubin and Cellina's *Differential Inclusion*, it is claimed that the property is equivalent to

given any generalized sequence $x_\mu \to x_0$ and any $y_0\in F(x_0)$, there exists a generalized sequence $y_\mu \in F(x_\mu)$ such that $y_\mu\to y_0$.

I assume that they meant **net** when they said generalized sequence.

I don't find this claim to be trivial at all. In fact, I found a proof of this from a different source that I believe to be invalid.

Let $(x_\mu)_{\mu\in \mathcal I}$ be a net converging to $x_0$, the aforementioned proof construct a net $y:D\to Y$ from a directed set $D\subset \mathcal I \times \mathcal N(y_0)$ defined by $$ D:=\{(\mu,V) : F(x_\mu)\cap V \neq\emptyset \} $$ by choosing $$ y_{\mu,V} \in F(x_\mu)\cap V. $$ It is not hard to verify that $(y_{\mu,V})_{(\mu,V)\to D}$ converges to $y_0$. However, their last step of finding a map $\sigma:\mathcal I \to D$ to compose with $y$ seems invalid, since they let $\sigma$ be a arbitrary section of $\pi:D\to \mathcal I$, where $\pi$ is the projection $\pi(\mu,V)=\mu$.

However, I have looked around and found an old paper by Frolík, *Concerning topological convergence of sets*, from 1960. It contains a statement that implies a weaker result (but seems more reasonable to me) that could be paraphrased as follows.

Let $(M_\mu)_{\mu\in \mathcal I}$ be a net in $2^Y$. Then $y_0$ is in $\liminf M_\mu$ if and only if there exists a net $\pi:\mathcal I'\to \mathcal I$ which is

residualin $\mathcal I$ and points $y_\lambda \in M_{\pi(\lambda)}$ such that $(y_\lambda)_{\lambda\in\mathcal I'} \to y_0$.

The proof is almost identical to the previous one but he stopped by letting $\mathcal I'=D$, defined as in the above proof, and concluded the theorem without having to invoke a section $\sigma$ of $\pi$.

Question: Is the claim by Aubin and Cellina true? If it is, then where can I find a valid proof of it?