As Terence tao explains we have the following conclusion that if we consider the empty set as subset of the extended reals. We have the following result:

We know that $sup(E) \geq inf(E)$. This would imply that $-\infty \geq +\infty$. This is non-sense based on the ordering on the extended reals. Why do we get this result ? Is there anything preventing us from getting this result ?

[![enter image description here](https://i.stack.imgur.com/1exHE.png)](https://i.stack.imgur.com/1exHE.png)