Infimum and supremum of the empty set Let $E$ be an empty set. Then, $\sup(E) = -\infty$ and $\inf(E)=+\infty$. I thought it is only meaningful to talk about $\inf(E)$ and $\sup(E)$ if $E$ is non-empty and bounded? 
Thank you. 

If $A\subset B\subset \mathbb R$, then lower bound of $B\leq $ lower bound of $A\leq$ upper bound of $B$, and lower bound of $B\leq$ the upper bound of $A\leq$ upper bound of $B$.
Now, consider  $p\in \mathbb Z$, where  $\mathbb Z$ is the set of +ve and -ve integers. Since $\emptyset \subset \{p\}$, so  lower bound of $\{p\}\leq$ lower bound of $\emptyset\leq$  upper bound of $\{p\}$, i.e. a lower bound of $\emptyset$ is $p\in \mathbb Z$. Hence, the greatest lower bound of $\emptyset$ is $\infty$.
Similarly, the least upper bound of $\emptyset$ is $-\infty$.
Notice that, $\emptyset$ has a nice/absurd property that it is a subset of any set, so it runs over the real line and so its upper and lower bound also run over the real line. While a nonempty set is fixed in its place, so it becomes unable to play such a game what $\emptyset$ plays.
