I am doing a proof of a Lemma that isn't in a book.
Let $X$ a Banach space and $\emptyset\not=S\subset X$ closed of $X$. Let $f$ be a lower semicontinuous function bounded below in $S$.
I have that $\displaystyle\lim_{\alpha\to 0^+} \mathop{diameter}(Sl(-f,S,\alpha))=0$, where $Sl(-f,S,\alpha)=\{x\in S\;:\, f(x)<\displaystyle\inf_{z\in S}\{f(z)\}+\alpha\}$.
I need to prove that:
$\exists x\in S$ such that $f(s)=\inf_{z\in S}\{f(z)\}$ and given a sequence $(x_n)\subset S$ such that $f(x_n)\to f(x)$, then $\|x_n-x\|\to 0$.
Thanks in advance.