Supremum and Infimum Equivalent Statements

# Equivalent Statements for The Supremum/Infimum of a Bounded Set

We will now look at some equivalent statements regarding the supremum and infimum of a set. When proving whether a number is the supremum of a bounded above set or the infimum of a bounded below set, these equivalent statements will come in handy.

Let $S$ be a set that is bounded above with $u = \sup S$. The following are equivalent:

• 1) If $v$ is any upper bound of $S$ then $u ≤ v$.
• 2) If $u' < u$, then $u'$ is not an upper bound of $S$.
• 3) If $z < u$ then there exists an $x' \in S$ such that $z < x'$.
• 4) If $\forall \epsilon > 0$ then there exists an $x_{\epsilon} \in S$ such that $u - \epsilon < x_{\epsilon}$.

The following set of equivalent statements pertain to the infimum of a set that is bounded below. They are analogous to the equivalent statements above regarding the supremum of a set that is bounded above.

Let $S$ be a set that is bounded below with $w = \inf S$. The following are equivalent:

• 1) If $t$ is any lower bound of $S$ then $t ≤ w$.
• 2) If $w < w'$, then $w'$ is not a lower bound of $S$.
• 3) If $w < z$ then there exists an $x' \in S$ such that $x' < z$.
• 4) If $\forall \epsilon > 0$ then there exists an $x_{\epsilon} \in S$ such that $x_{\epsilon} < w + \epsilon$.