1.1 Suprema and Infima
Hopefully by now you’re getting more familiar with the ideas of set bounds, especially the idea of suprema/infima! It also turns out that there’s an alternative characterisation of suprema and infima which can be very useful, especially if the members of a set aren’t indexed by natural numbers. Technically, this is something that came up in last week’s lectures, but is more relevant to this week’s problem sheet (see Tutorial Question 1).
Let \(S\subseteq\mathbb{R}\). Then a number \(T\in\mathbb{R}\) is the supremum of \(S\), denoted \(\sup(S)\) if: \[\forall \epsilon > 0, \exists s \in S\; \text{such that} \; s > T - \epsilon.\]
Let \(S\subseteq\mathbb{R}\). Then a number \(t\in\mathbb{R}\) is the infimum of \(S\), denoted \(\inf(S)\) if: \[\forall \epsilon > 0, \exists s \in S\; \text{such that} \; s < t + \epsilon.\]
As an example, take the set \(S = (-1,2] = \lbrace x \, \lvert\, -1 < x \leq 2\rbrace\), and fix some \(\epsilon > 0\). Then, if we take \(s_1 = 2 - \epsilon/2\) and \(s_2 = -1 + \epsilon/2\), we see that
- \(s_1\) and \(s_2\) are in the set \(S\),
- \(s_1 > 2 - \epsilon\), and
- \(s_2 < -1 + \epsilon\).
Hence, as \(\epsilon\) was arbitrary, the alternative characterisation of suprema and infima says that \(\sup(S) = 2\) and \(\inf(S) = -1\).