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Math and science::Analysis::Tao::05. The real numbers

Supremum

Let \( E \) be a subset of \( \mathbb{R} \). If \( E \) is not empty and has some upper bound, we define \( sup(E) \) to be [...].

We introduce two new symbols, \( +\infty, -\infty \), to deal with two special cases. If \( E \) is non-empty and [...], we set \( sup(E) := +\infty \); if \( E \) [...], we set \( sup(E) := -\infty \).

We refer to \( sup(E) \) as the supremum of E, and denote it as sup E.