Meaning of symbols like $\inf\limits_{\epsilon>0}$

Question

I am very confused at the precise definition of the following symbols. A reference or explanation would be great.

$$\Large\inf\limits_{\epsilon>0}\qquad \sup\limits_{\epsilon>0}$$

Answer 1

For example, for $f:\mathbb R\to\mathbb R$ we have $$ \inf_{\epsilon > 0} f(\epsilon) = \inf \{ f(\epsilon) \mid \epsilon > 0 \}, $$ where $\inf$ means the infimum of a real subset, that is, the greatest lower bound of that subset. $\sup$ means the supremum, that is the least upper bound.

Answer 2

another definition of sup and inf: Let say X is set

$$supX=\alpha \\ 1.\forall x\epsilon X,\quad x\le \alpha \\ 2.\quad \varepsilon >0,\quad x>\alpha -\varepsilon $$

and

$$\\ infX=\beta \\ 1.\forall x\epsilon X,\quad x\ge \beta \\ 2.\quad \varepsilon >0,\quad x<\beta +\varepsilon $$