1.1 Suprema and Infima

There’s still a bit of material to cover regarding the supremum and infimum of a set. To begin, we re-cover the definitions from last week.

Definition 1.1: (Supremum)

Let \(S \in \mathbb{R}\). A number \(T \in \mathbb{R}\) is said to be the supremum of \(S\) if it is an upper bound for \(S\), and for any other upper bound \(M\), \(T \leq M\). Here, we write \(T = \sup(S)\).

Definition 1.2: (Infimum)

Let \(S \in \mathbb{R}\). A number \(t \in \mathbb{R}\) is said to be the infimum of \(S\) if it is a lower bound for \(S\), and for any other lower bound \(m\), \(t\geq m\). Here, we write \(t = \inf(S)\).

There’s also two results from last week’s notes that you didn’t reach in lectures either, so we (re)state these below too.

Completeness Axiom:

Every non-empty set \(S\) in \(\mathbb{R}\) that is bounded above has a supremum.

Proposition 1.1: (Archimedian Postulate)

We have that \(\forall x \in \mathbb{R}, \exists N \in \mathbb{N}\) such that \(N > x.\) In other words, the set of natural numbers \(\mathbb{N}\) is unbounded above.

As mentioned in last week’s notes, the completeness axiom assumes that there are no ‘gaps’ in the real number line (and allows us to solve equations such as \(x^2-2=0\), for example.) It’s also worth mentioning how useful Archimedes’ Postulate is — this is usually the result you will contradict when showing a set is not bounded.