The "min" operator often appears in delta - epsilon limit proofs.
What I can find while looking at limit proofs examples is something like this:
the teacher says "if $\delta<a$ AND $\delta<b$, then the implication defining a limit holds".
and after that, he adds: so lets choose $\delta = \min\{a, b\}$.
What I understand is that choosing $\delta = \min \{a, b\}$ guarantees that "$\delta < a$ AND $\delta < b$".
But I do not understand how the first guarantees the second.
I may be wrong in thinking that the first guarantees the second.
And my mistake may come from the fact I do not understand what means the min operator.
This is why I asked this question as to the precise meaning of "min".
Note : which key word should I enter on a search engine to get informations about this " min" operator? I guess" min" wouldn't work.
The usual definition of $\operatorname{min}$ is as follows: $$\operatorname{min}(x,y) =\begin{cases}x &\text{if }x\leq y \\y &\text{if } y\leq x\end{cases}$$ where the two cases cover every possibility (by totality of the order) and only overlap when $x=y$ (by antisymmetry). You show, by examining the two cases and using transitivity, that $\min(x,y)\leq x$ and $\min(x,y)\leq y$.
In fact, you can characterize $\min$ via this property as well: $\min(x,y)$ is an element of the set $\{x,y\}$ that is less than or equal to every other element. Antisymmetry of the order implies that this definition gives a at most one answer for any set and one may, via an inductive argument, show that such an element exists for every finite set.
One may also, a bit more sophisticatedly, note that $\min(x,y)$ is the greatest lower bound (a.k.a. the infimum) to the set $\{x,y\}$ meaning that it is a lower bound (i.e. is less or equal to both $x$ and $y$) and that every other lower bound is less than or equal to the minimum. This is sometimes handy to know when there's a possibility that infinite sets will show up - and this definition applies to some more general mathematical objects as well.
Note that you do not get that $\min(x,y) < x$ or $\min(x,y) < y$ in general - but this is just something to keep in mind - it's rarely a hard problem to fix while proving things about limits.