Lecture - Analysis MT22, III

What is the completeness axiom for a set $S$ that is a non-empty subset of $\mathbb{R}$?

If $S$ is bounded above, then $S$ has a supremum $\sup S \in \mathbb{R}$.

What’s a less fancy name for the supremum?

The least upper bound.

What does it mean for $b \in \mathbb{R}$ to be an upper bound for $S$?

$\forall x \in S, b \ge x$

What does it mean for $b \in \mathbb{R}$ to be a lower bound for $S$?:: $\forall x \in S, b \le x$

What does it mean for a set $S$ to be bounded?

It is bounded above and below.

What is a supremum of a set $S$?

The smallest upper bound for $S$.

What is $\sup [-1, \infty)$?

There isn’t one.

What is $\sup [-1, 2)$?

\[2\]

True or false: $\sup S \in S$

Not always.

What’s a less fancy name for the infimum?

The greatest lower bound.

What is true about $\sup S$ for $S \ne \emptyset$ and $S \subseteq T \subseteq \mathbb{R}$?

\[\sup S \le \sup T\]