My professor defined lower semicontinuous envelope$f_{*}(x):=\inf \left\{\liminf _{n \rightarrow+\infty} f\left(x_n\right): x_n \rightarrow x\right\}$.
his proof fixed $x_0$ and $x_k\rightarrow x_0$by definition$\exists y_k\quad d(y_k,x_k)<\epsilon_k \quad \text{s.t}\quad f_{*}(x_k)\geq f(y_k)-\epsilon_k$
so $\liminf f_{*}(x_k)\geq f(y_k)-\epsilon \geq f_{*}(x_0)$
My problem I don't understand when he says by definition , I mean it is right what he wrote but I find that he did too unnecessary work because the real definition of $inf$ should be$\exists y_k\rightarrow x_k : \liminf_{y_k\rightarrow x_k} f(y_k)\leq f_{*}(x_k)-\epsilon_k$
so applying $\liminf$ to both member I have thesis
Is it correct even this last version ? I think the 2 proofs are very similar but the difference is that I think my professor also explicited (implicitly) $\liminf_{y_k\rightarrow x_k} f(y_k)\leq f_{*}(x_k)-\epsilon_k$ and then he applyed again \liminf so for this I think too much work