Suppose
$\emptyset\neq E\subset F \subset \mathbb R$
Prove that $\inf E \ge \inf F$ and $\sup E \le \sup F$.
The second part I kind of understand. If $E$ is a subset of $F$ and $\sup F$ is an upper bound for $F$ then $\sup F$ is going to be an upper bound for $E$.
Therefore $\sup E \le \sup F$
But the infinitum part I am not sure how to show.
The infimum of $F$ is the greatest element of $\mathbb{R}$ that is less or equal to all elements of $F$ so, since $E \subset F$ is also less or equal to all elements of $E$. If there are no elements in $F$ less than all elements of $E$ than it is also the infimum of $E$, else it is less than inf$(E)$.