Inequality for infimum over intersection of sets

Question

Let $f$ be a function $f: \mathbb{R}^n \to \mathbb{R}$, and let $A,B \subset \mathbb{R}^n$.

Is the following in general true? $$\inf_{x \in A} f(x) \leq \inf_{x \in A\cap B} f(x)$$

Answer 1

I suppose that for "taking values" you mean the domain of the function, otherwise the notion is not well defined in general (you may solve the problem defining orders on vectors in the sets $f(A),f(B)$ you are considering in the codomain.)

It comes straight from the definition of infimum. As in one comment is mentioned, the infimum of a (weakly) bigger set is for sure (weakly) smaller than the infimum of the other set. Since $A \cap B \subseteq A$, the claim follows.

Recall: The infimum of a set $S$ is defined as the maximum of the lower bounds of the set $A$.