Definition of lim inf?

Question

I would like to know what is the precise definition of : $$\lim _{x\rightarrow a} \inf f(x) $$ when $f : R^n \rightarrow R$ in my course it is written that : $$\lim _{x\rightarrow a} \inf f(x) =\sup_{\epsilon>0}\inf_{x\neq a, \mid \mid x-a\mid \mid <\epsilon}f(x)$$ But I don't really understand what represents $\sup$ here ? We take the $\sup$ of what ?

Thanks,

Answer 1

Consider the set $A(\epsilon) =\{x \in \mathbb R^n \mid \Vert x-a \Vert <\epsilon\}\setminus \{a\}$.

You can define $g(\epsilon)=\inf\limits_{x \in A(\epsilon)} f(x)$. $g$ depends on $\epsilon$.

Then $\liminf\limits_{x \to a} f(x) = \sup\limits_{\epsilon >0} g(\epsilon)$.