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. 

We can prove it by contradiction. Let $S = \varnothing$ be the empty set and let $S$ does not have any upper bound i.e. there exists no real number $M$ such that $M$ is greater than or equal to every $x \in S$, which implies there exist at least one element $x \in S$ such that $x > M$, which is a contradiction since $S$ is an empty set and has no elements.
Similarly if $S$ does not have any lower bound then there exist no real number $m$ which is smaller than or equal to every element of $S$, implies there exist at least one element $x \in S$ such that $x < m$, again a contradiction since $S$ is an empty set and has no elements.
Hence our assumption is wrong and $S$ has both upper and lower bounds in fact every real number is an upper bound and every real number is a lower bound of an empty set $S$.
But an empty set has no least upper bound since we don't have any smallest real number, similarly an empty set has no greatest lower bound since there exists no greatest real number.
