1.3 Sets and Bounds
1.3.1 Upper and Lower Bounds
The final topic we are going to re-cover here involves sets of real numbers, and deciding on whether we can ‘trap’ these sets between an upper bound — which no member of the set can lie above — and a lower bound — which no member of the set can lie below. To do this requires a few defintions, which are presented below:
Definition 1.7: (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 1.8: (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 1.9: (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 \(x \in S\), \(m\leq x \leq M\).
1.3.2 Suprema and Infima
Think about upper bounds for a moment: if we have one, we could ask if there is a smaller number that 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, analogously for lower bounds, infima:
Definition 1.10: (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 1.11: (Infimum)
Finally, we state an alternative characterisation of the supremum/infimum of a set.
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 \(t = \inf(S)\).
Proposition 1.7:
Let \(S\subseteq\mathbb{R}\). Then a number \(T\in\mathbb{R}\) is the supremum of \(S\), denoted \(\sup(S)\) if: \[\forall \epsilon > 0, \exists s \in S\; \text{such that} \; s > T - \epsilon.\]
Proposition 1.8:
Let \(S\subseteq\mathbb{R}\). Then a number \(t\in\mathbb{R}\) is the infimum of \(S\), denoted \(\inf(S)\) if: \[\forall \epsilon > 0, \exists s \in S\; \text{such that} \; s < t + \epsilon.\]