1.3 Set Bounds

You might be interested to know that at this stage, despite the fact we have specified 14 axioms, these still aren’t unique to the real numbers! For example, the exact same axioms also apply to the set of rational numbers \(\mathbb{Q}\). Luckily, we only need one more axiom to complete our description of the real numbers. Unfortunately, there are a few definitions we need first…

Definition 1.1: (Upper Bound)

Let \(S \subseteq \mathbb{R}\). Then \(M \in \mathbb{R}\) is an upper bound for \(S\) if for all \(x \in S\), \(x \leq M\). In this case, we say \(S\) is bounded above.

Definition 1.2: (Lower Bound)

Let \(S \subseteq \mathbb{R}\). Then \(m \in \mathbb{R}\) is a lower bound for \(S\) if for all \(x \in S\), \(x \geq m\). In this case, we say that \(S\) is bounded below.

Definition 1.3: (Bounded Set)

A set \(S\) is bounded if it is both bounded above and below. Equivalently, \(S\) is bounded if there exists \(m, M \in \mathbb{R}\) such that for all \(x \in S\), \(m\leq x \leq M\).

We can go one step further with the definition of a bounded set. Namely, we can say that \(S\) is bounded if \(\exists M^{*}\geq0\) such that \(\lvert x \rvert \leq M\) for all members \(x\) of \(S\). The link here is that \(M^{*} = \max\left\lbrace \lvert m \rvert, \lvert M \rvert \right\rbrace\).

Thinking of upper bounds for a moment, if we have one, we could ask if there is a smaller number which also bounds the set from above. You might also be tempted to ask what the ‘best’ upper bound on a set could be, such that no smaller number will bound the set from above. This leads to the ideas of suprema and infima:

Definition 1.4: (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 1.5: (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 \(t = \inf(S)\).

For example, if we consider the set \(S = (-1,2] = \lbrace x \lvert -1<x\leq2\rbrace\), we can see that possible upper and lower bounds are \(M = 3\) and \(m = -2\) respectively, so the set is bounded. Its supremum and infimum are \(\sup(S) = 2\) and \(\inf(S) = -1\). However, note that the supremum lies inside \(S\), whereas the infimum does not lie inside \(S\). This also tells us that the maximum element of \(S\) is \(2\), whereas \(S\) has no minimum element!

1.3.1 The Completeness Axiom

Finally, we are ready to state the required \(15^{th}\) axiom! As the title suggests, this is known as the Completeness Axiom.

Completeness Axiom:

Every non-empty set \(S\) in \(\mathbb{R}\) that is bounded above has a supremum³.

Loosely, this axiom ensures that there are no ‘gaps’ in the real number line. For some more (precise) information, see this link.

1.3.2 The Archimedian Postulate

To finish, we mention one result which will become very useful when studying sequences in the next few weeks. This is the Archimedian Postulate, and says that the set of natural numbers is unbounded above. In maths terms:

Proposition 1.1: (Archimedian Postulate)

We have that \(\forall x \in \mathbb{R}, \exists N \in \mathbb{N}\) such that \(N > x.\)

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3. In the lecture notes, it also states that ‘Every non-empty set of real numbers that is bounded below has an infimum.’ But you can deduce this from the supremum result by considering the set \(-S:=\lbrace -x \lvert x \in S\rbrace.\)↩︎