We are given a collection of models $\mathcal{F}$ from which particular, but arbitrarily chosen models $f$ are sampled and a function $\text{err}(f')$ returns the error in classification of the model $f'$.
Now, we define a model $f^* := \text{argmin}_{ f \in \mathcal{F}} \text{err}(f)$ (the model in $\mathcal{F}$ that produces the minimum error).
Question
Is the following statement correct?
$$ \inf_{f \in \mathcal{F}} \text{ err}(f) = \text{err}(f^*) $$
Additionally, why use infimum instead of minimum?
That's it. Thanks!