Let $$E$$ be a subset of $$\mathbb{R}$$. If $$E$$ is not empty and has some upper bound, we define $$sup(E)$$ to be [...].
We introduce two new symbols, $$+\infty, -\infty$$, to deal with two special cases. If $$E$$ is non-empty and [...], we set $$sup(E) := +\infty$$; if $$E$$ [...], we set $$sup(E) := -\infty$$.
We refer to $$sup(E)$$ as the supremum of E, and denote it as sup E.