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Math and science::Analysis::Tao::06. Limits of sequences

Supremum of sets of extended reals

We wish to transitioning from sets of reals to sets of extended reals. We define the supremum and infimum of sets of extended reals like so:

Let E be a subset of \( \mathbb{R}^* \). Then we define the supremum of \( sup(E) \) or least upper bound of \( E \) by the following rule.

  1. If \( E \) is contained in \( \mathbb{R} \) (i.e., \( +\infty \) and \( -\infty \) are not elements of E), then we let sup(E) be the supremum already defined for sets of reals.
  2. If \( E \) contains \( +\infty \), then we set \( sup(E) \) like so: \( sup(E) := +\infty \).
  3. If E does not contain \( +\infty \) but does contain \( -\infty \), then we set \( sup(E) := sup(E\setminus \{-\infty\}) \) (which is a subset of \( \mathbb{R} \) and thus falls under case 1).

We also (some what lazily, or craftily) define the infimum \( inf(E) \) of \( E \) (also known as the greatest lower bound of \( E \) by the formula:

\[ inf(E) := -sup(-E) \]

Where \( -E \) is the set \( -E := \{-x : x \in E\} \).