# Lecture - Analysis MT22, III

> Source: https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/lectures/3/ · Updated: 2022-10-23 · Tags: uni, lecture

- [Course - Analysis MT22](https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/)

### Flashcards
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 $S \subseteq T \subseteq \mathbb{R}$?::
$$
\sup S \le \sup T
$$

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