How to show Sup over bigger set is bigger and Inf over bigger set is lesser?

Question

Let $A,B \in \mathbb{R}^n$ be two subset such that $A \subseteq B$. Also, let $f:\mathbb{R}^n \rightarrow \mathbb{R}$ be a real-valued function.

I always use

$$\sup_Af(x) \leq \sup_Bf(x)$$

That perfectly makes sense to me, but is there any proof for the above inequality?

Also for the following

$$\inf_Bf(x) \leq \inf_Af(x)$$

Answer 1

Hint: $\sup_{x\in B}f(x)$ is an upper bound of the set $\{f(x)\,|\,x\in A\}$.

Answer 2

Let $x\in A$. Then in particular $x\in B$ and $f(x)\geq \inf_{x\in B}f(x)$. So $\inf_{x\in B}f(x)$ is a lower bound for the values of $f(x)$ on $x\in A$. In particular $\inf_{x\in A}f(x)\geq \inf_{x\in B}f(x)$.